:: NDIFF_4 semantic presentation

REAL is non empty V49() V160() V161() V162() V166() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() Element of K6(REAL)
K6(REAL) is set
REAL * is functional FinSequence-membered FinSequenceSet of REAL
K7(NAT,(REAL *)) is set
K6(K7(NAT,(REAL *))) is set
COMPLEX is non empty V49() V160() V166() set
K7(REAL,REAL) is complex-valued ext-real-valued real-valued set
K6(K7(REAL,REAL)) is set
omega is non empty epsilon-transitive epsilon-connected ordinal V160() V161() V162() V163() V164() V165() V166() set
K6(omega) is set
K6(NAT) is set
K7(NAT,REAL) is complex-valued ext-real-valued real-valued set
K6(K7(NAT,REAL)) is set
K224() is non empty set
K7(K224(),K224()) is set
K7(K7(K224(),K224()),K224()) is set
K6(K7(K7(K224(),K224()),K224())) is set
K7(REAL,K224()) is set
K7(K7(REAL,K224()),K224()) is set
K6(K7(K7(REAL,K224()),K224())) is set
K230() is RLSStruct
the carrier of K230() is set
K6( the carrier of K230()) is set
K234() is Element of K6( the carrier of K230())
K7(K234(),K234()) is set
K7(K7(K234(),K234()),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(K234(),K234()),REAL)) is set
ExtREAL is non empty V161() set
RAT is non empty V49() V160() V161() V162() V163() V166() set
INT is non empty V49() V160() V161() V162() V163() V164() V166() set
K7(COMPLEX,COMPLEX) is complex-valued set
K6(K7(COMPLEX,COMPLEX)) is set
K7(K7(COMPLEX,COMPLEX),COMPLEX) is complex-valued set
K6(K7(K7(COMPLEX,COMPLEX),COMPLEX)) is set
K7(K7(REAL,REAL),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(REAL,REAL),REAL)) is set
K7(RAT,RAT) is RAT -valued complex-valued ext-real-valued real-valued set
K6(K7(RAT,RAT)) is set
K7(K7(RAT,RAT),RAT) is RAT -valued complex-valued ext-real-valued real-valued set
K6(K7(K7(RAT,RAT),RAT)) is set
K7(INT,INT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued set
K6(K7(INT,INT)) is set
K7(K7(INT,INT),INT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued set
K6(K7(K7(INT,INT),INT)) is set
K7(NAT,NAT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K7(K7(NAT,NAT),NAT) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7(K7(NAT,NAT),NAT)) is set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
K7(1,1) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7(1,1)) is set
K7(K7(1,1),1) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K6(K7(K7(1,1),1)) is set
K7(K7(1,1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(1,1),REAL)) is set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
K7(2,2) is RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K7(K7(2,2),REAL) is complex-valued ext-real-valued real-valued set
K6(K7(K7(2,2),REAL)) is set
TOP-REAL 2 is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V202() L20()
the carrier of (TOP-REAL 2) is non empty set
REAL 1 is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
1 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL 1),(REAL 1)) is set
K6(K7((REAL 1),(REAL 1))) is set
REAL-NS 1 is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS 1) is non empty non trivial set
R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() NORMSTR
BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))))
R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS 1)) is non empty functional Element of K6( the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS 1)))
K284( the carrier of (REAL-NS 1),(REAL-NS 1)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)
FuncZero ( the carrier of (REAL-NS 1),(REAL-NS 1)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))
0. (REAL-NS 1) is V68( REAL-NS 1) Element of the carrier of (REAL-NS 1)
the ZeroF of (REAL-NS 1) is Element of the carrier of (REAL-NS 1)
the carrier of (REAL-NS 1) --> (0. (REAL-NS 1)) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))) is set
FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS 1)) is Relation-like K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))))
K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))))) is set
FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS 1)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1))),(FuncZero ( the carrier of (REAL-NS 1),(REAL-NS 1))),(FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS 1))),(FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS 1))) #) is strict RLSStruct
the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS 1)) is non empty set
K6( the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS 1))) is set
K233(K284( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS 1),(REAL-NS 1))
K231(K284( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS 1))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
K232(K284( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),(LinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS 1))),K233(K284( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),K231(K284( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))),K232(K284( the carrier of (REAL-NS 1),(REAL-NS 1)),(LinearOperators ((REAL-NS 1),(REAL-NS 1)))) #) is V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K233((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Element of BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))
K231((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
K232((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))))) is set
BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS 1)) is non empty Relation-like BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))),K233((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),K231((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),K232((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS 1))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1)))),(BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS 1))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS 1))) is non empty set
TOP-REAL 1 is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V202() L20()
the carrier of (TOP-REAL 1) is non empty set
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V132() V159() V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() Element of NAT
- 1 is V11() real ext-real non positive Element of REAL
|.0.| is V11() real ext-real V159() Element of REAL
sqrreal is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
proj (1,1) is non empty Relation-like REAL 1 -defined REAL 1 -defined REAL -valued Function-like one-to-one total total quasi_total quasi_total onto bijective complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
K7((REAL 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL 1),REAL)) is set
(proj (1,1)) " is Relation-like Function-like one-to-one set
K7(REAL,(REAL 1)) is set
K6(K7(REAL,(REAL 1))) is set
dom ((proj (1,1)) ") is set
rng ((proj (1,1)) ") is set
- 1 is V11() real ext-real non positive Element of REAL
NAT --> 0 is non empty T-Sequence-like Relation-like NAT -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of K6(K7(NAT,NAT))
K6(K7(NAT,NAT)) is set
[#] REAL is V160() V161() V162() closed open Element of K6(REAL)
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
m - x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
K714(REAL,REAL,(REAL n),(REAL n),m,x0) is Relation-like REAL /\ REAL -defined K640((K635((REAL n)) /\ K635((REAL n)))) -valued Function-like V239() V240() V241() Element of K6(K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n))))))
REAL /\ REAL is V160() V161() V162() set
K635((REAL n)) is set
K635((REAL n)) /\ K635((REAL n)) is set
K640((K635((REAL n)) /\ K635((REAL n)))) is functional V233() V234() V235() set
K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n))))) is set
K6(K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n)))))) is set
- x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
m + (- x0) is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
K708(REAL,REAL,(REAL n),(REAL n),m,(- x0)) is Relation-like REAL /\ REAL -defined K640((K635((REAL n)) /\ K635((REAL n)))) -valued Function-like V239() V240() V241() Element of K6(K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n))))))
dom (m - x0) is V160() V161() V162() Element of K6(REAL)
dom m is V160() V161() V162() Element of K6(REAL)
dom x0 is V160() V161() V162() Element of K6(REAL)
(dom m) /\ (dom x0) is V160() V161() V162() Element of K6(REAL)
dom (m + (- x0)) is V160() V161() V162() Element of K6(REAL)
dom (- x0) is V160() V161() V162() Element of K6(REAL)
(dom m) /\ (dom (- x0)) is V160() V161() V162() Element of K6(REAL)
S is set
x0 . S is Relation-like Function-like complex-valued ext-real-valued real-valued set
x0 /. S is Relation-like Function-like complex-valued ext-real-valued real-valued V56(n) Element of REAL n
(- x0) . S is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- x0) /. S is Relation-like Function-like complex-valued ext-real-valued real-valued V56(n) Element of REAL n
(m - x0) . S is Relation-like Function-like complex-valued ext-real-valued real-valued set
m . S is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m . S) - (x0 . S) is Relation-like Function-like complex-valued ext-real-valued real-valued set
- (x0 . S) is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) (x0 . S) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m . S) + (- (x0 . S)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m . S) + ((- x0) . S) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m + (- x0)) . S is Relation-like Function-like complex-valued ext-real-valued real-valued set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
S is V11() real ext-real set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 is V11() real ext-real set
S is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
diff (S,x0) is Element of the carrier of (REAL-NS n)
g is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
S is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
diff (g,x0) is Element of the carrier of (REAL-NS n)
N1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
diff (N1,x0) is Element of the carrier of (REAL-NS n)
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
S is V11() real ext-real set
(n,m,S) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
diff (x0,S) is Element of the carrier of (REAL-NS n)
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
diff (T,S) is Element of the carrier of (REAL-NS n)
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
n is set
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
dom x0 is V160() V161() V162() Element of K6(REAL)
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is V160() V161() V162() open Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom x0 is V160() V161() V162() Element of K6(REAL)
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
S is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom S is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
x0 | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
S | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T is V11() real ext-real Element of REAL
T is V11() real ext-real Element of REAL
T is V11() real ext-real Element of REAL
S is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T is V11() real ext-real Element of REAL
S | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
x0 | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is V160() V161() V162() Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
S is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom S is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
x0 | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
S | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 is set
S is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom S is V160() V161() V162() Element of K6(REAL)
T is set
T is set
T is V11() real ext-real Element of REAL
S . T is Relation-like Function-like complex-valued ext-real-valued real-valued set
(n,m,T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
T is V11() real ext-real Element of REAL
S . T is Relation-like Function-like complex-valued ext-real-valued real-valued set
(n,m,T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
S is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom S is V160() V161() V162() Element of K6(REAL)
T is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom T is V160() V161() V162() Element of K6(REAL)
g is V11() real ext-real Element of REAL
S . g is Relation-like Function-like complex-valued ext-real-valued real-valued set
(n,m,g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
T . g is Relation-like Function-like complex-valued ext-real-valued real-valued set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
0* n is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of n -tuples_on REAL
m is V160() V161() V162() open Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom x0 is V160() V161() V162() Element of K6(REAL)
rng x0 is functional V233() V234() V235() Element of K6((REAL n))
K6((REAL n)) is set
(n,x0,m) is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
S is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
{S} is non empty functional V233() set
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T `| m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
g is V11() real ext-real Element of REAL
(T `| m) /. g is Element of the carrier of (REAL-NS n)
g is V11() real ext-real Element of REAL
T | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
x0 | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
g is V11() real ext-real Element of REAL
(T `| m) /. g is Element of the carrier of (REAL-NS n)
dom (T `| m) is V160() V161() V162() Element of K6(REAL)
(T `| m) . g is set
diff (T,g) is Element of the carrier of (REAL-NS n)
(n,x0,m) . g is Relation-like Function-like complex-valued ext-real-valued real-valued set
(n,x0,g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
dom (n,x0,m) is V160() V161() V162() Element of K6(REAL)
(n,x0,m) /. g is Relation-like Function-like complex-valued ext-real-valued real-valued V56(n) Element of REAL n
g is V11() real ext-real Element of REAL
(n,x0,m) /. g is Relation-like Function-like complex-valued ext-real-valued real-valued V56(n) Element of REAL n
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is V11() real ext-real set
{m} is non empty V160() V161() V162() set
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom x0 is V160() V161() V162() Element of K6(REAL)
(n,x0,m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
S is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T is V160() V161() V162() open Neighbourhood of m
g is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
g " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
N1 is non empty Relation-like NAT -defined REAL -valued Function-like constant total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
rng N1 is trivial V160() V161() V162() Element of K6(REAL)
g + N1 is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
rng (g + N1) is V160() V161() V162() Element of K6(REAL)
S /* (g + N1) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
K7(NAT, the carrier of (REAL-NS n)) is set
K6(K7(NAT, the carrier of (REAL-NS n))) is set
S /* N1 is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
(S /* (g + N1)) - (S /* N1) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
(g ") (#) ((S /* (g + N1)) - (S /* N1)) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
lim ((g ") (#) ((S /* (g + N1)) - (S /* N1))) is Element of the carrier of (REAL-NS n)
diff (S,m) is Element of the carrier of (REAL-NS n)
n is V11() real ext-real Element of REAL
m is V11() real ext-real Element of REAL
x0 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL x0 is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
x0 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL x0)) is set
K6(K7(REAL,(REAL x0))) is set
S is Relation-like REAL -defined REAL x0 -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL x0)))
m (#) S is Relation-like REAL -defined REAL x0 -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL x0)))
K677(REAL,(REAL x0),S,m) is Relation-like REAL -defined K640(K635((REAL x0))) -valued Function-like V239() V240() V241() Element of K6(K7(REAL,K640(K635((REAL x0)))))
K635((REAL x0)) is set
K640(K635((REAL x0))) is functional V233() V234() V235() set
K7(REAL,K640(K635((REAL x0)))) is set
K6(K7(REAL,K640(K635((REAL x0))))) is set
(x0,(m (#) S),n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(x0) FinSequence-like Element of REAL x0
(x0,S,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(x0) FinSequence-like Element of REAL x0
m * (x0,S,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(x0) FinSequence-like Element of REAL x0
K395(m) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (m,(id REAL)) is set
(x0,S,n) (#) K395(m) is Relation-like Function-like complex-valued ext-real-valued real-valued set
REAL-NS x0 is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS x0) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS x0)) is set
K6(K7(REAL, the carrier of (REAL-NS x0))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS x0) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS x0)))
m (#) T is Relation-like REAL -defined the carrier of (REAL-NS x0) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS x0)))
diff ((m (#) T),n) is Element of the carrier of (REAL-NS x0)
diff (T,n) is Element of the carrier of (REAL-NS x0)
m * (diff (T,n)) is Element of the carrier of (REAL-NS x0)
the Mult of (REAL-NS x0) is Relation-like K7(REAL, the carrier of (REAL-NS x0)) -defined the carrier of (REAL-NS x0) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS x0)), the carrier of (REAL-NS x0)))
K7(K7(REAL, the carrier of (REAL-NS x0)), the carrier of (REAL-NS x0)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS x0)), the carrier of (REAL-NS x0))) is set
K466( the Mult of (REAL-NS x0),m,(diff (T,n))) is set
n is V11() real ext-real Element of REAL
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
- x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
(m,(- x0),n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(m,x0,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
- (m,x0,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
K446() is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(m,x0,n) (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) (m,x0,n) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K395(K38(1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (K38(1),(id REAL)) is set
(m,x0,n) (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) (#) x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
K677(REAL,(REAL m),x0,(- 1)) is Relation-like REAL -defined K640(K635((REAL m))) -valued Function-like V239() V240() V241() Element of K6(K7(REAL,K640(K635((REAL m)))))
K635((REAL m)) is set
K640(K635((REAL m))) is functional V233() V234() V235() set
K7(REAL,K640(K635((REAL m)))) is set
K6(K7(REAL,K640(K635((REAL m))))) is set
n is V11() real ext-real Element of REAL
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
(m,x0,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
S is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
x0 + S is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
K708(REAL,REAL,(REAL m),(REAL m),x0,S) is Relation-like REAL /\ REAL -defined K640((K635((REAL m)) /\ K635((REAL m)))) -valued Function-like V239() V240() V241() Element of K6(K7((REAL /\ REAL),K640((K635((REAL m)) /\ K635((REAL m))))))
REAL /\ REAL is V160() V161() V162() set
K635((REAL m)) is set
K635((REAL m)) /\ K635((REAL m)) is set
K640((K635((REAL m)) /\ K635((REAL m)))) is functional V233() V234() V235() set
K7((REAL /\ REAL),K640((K635((REAL m)) /\ K635((REAL m))))) is set
K6(K7((REAL /\ REAL),K640((K635((REAL m)) /\ K635((REAL m)))))) is set
(m,(x0 + S),n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(m,S,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(m,x0,n) + (m,S,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() .: ((m,x0,n),(m,S,n)) is set
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
g is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
T + g is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
diff ((T + g),n) is Element of the carrier of (REAL-NS m)
diff (T,n) is Element of the carrier of (REAL-NS m)
diff (g,n) is Element of the carrier of (REAL-NS m)
(diff (T,n)) + (diff (g,n)) is Element of the carrier of (REAL-NS m)
n is V11() real ext-real Element of REAL
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
(m,x0,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
S is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
x0 - S is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
K714(REAL,REAL,(REAL m),(REAL m),x0,S) is Relation-like REAL /\ REAL -defined K640((K635((REAL m)) /\ K635((REAL m)))) -valued Function-like V239() V240() V241() Element of K6(K7((REAL /\ REAL),K640((K635((REAL m)) /\ K635((REAL m))))))
REAL /\ REAL is V160() V161() V162() set
K635((REAL m)) is set
K635((REAL m)) /\ K635((REAL m)) is set
K640((K635((REAL m)) /\ K635((REAL m)))) is functional V233() V234() V235() set
K7((REAL /\ REAL),K640((K635((REAL m)) /\ K635((REAL m))))) is set
K6(K7((REAL /\ REAL),K640((K635((REAL m)) /\ K635((REAL m)))))) is set
(m,(x0 - S),n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(m,S,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(m,x0,n) - (m,S,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
K449() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K446() is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K448() * ((id REAL),K446()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K449() .: ((m,x0,n),(m,S,n)) is set
- (m,S,n) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
(m,S,n) (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) (m,S,n) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K395(K38(1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K450() [;] (K38(1),(id REAL)) is set
(m,S,n) (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m,x0,n) + (- (m,S,n)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: ((m,x0,n),(- (m,S,n))) is set
- S is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
x0 + (- S) is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
K708(REAL,REAL,(REAL m),(REAL m),x0,(- S)) is Relation-like REAL /\ REAL -defined K640((K635((REAL m)) /\ K635((REAL m)))) -valued Function-like V239() V240() V241() Element of K6(K7((REAL /\ REAL),K640((K635((REAL m)) /\ K635((REAL m))))))
(m,(- S),n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
- (m,S,n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
n is V11() real ext-real Element of REAL
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
x0 is V160() V161() V162() open Element of K6(REAL)
S is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
dom S is V160() V161() V162() Element of K6(REAL)
n (#) S is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
K677(REAL,(REAL m),S,n) is Relation-like REAL -defined K640(K635((REAL m))) -valued Function-like V239() V240() V241() Element of K6(K7(REAL,K640(K635((REAL m)))))
K635((REAL m)) is set
K640(K635((REAL m))) is functional V233() V234() V235() set
K7(REAL,K640(K635((REAL m)))) is set
K6(K7(REAL,K640(K635((REAL m))))) is set
(m,(n (#) S),x0) is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
dom (n (#) S) is V160() V161() V162() Element of K6(REAL)
T is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
n (#) T is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
dom (n (#) T) is V160() V161() V162() Element of K6(REAL)
dom T is V160() V161() V162() Element of K6(REAL)
g is V11() real ext-real Element of REAL
S | x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
T | x0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
(n (#) T) `| x0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
g is V11() real ext-real Element of REAL
((n (#) T) `| x0) . g is set
diff (T,g) is Element of the carrier of (REAL-NS m)
n * (diff (T,g)) is Element of the carrier of (REAL-NS m)
the Mult of (REAL-NS m) is Relation-like K7(REAL, the carrier of (REAL-NS m)) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m)))
K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m))) is set
K466( the Mult of (REAL-NS m),n,(diff (T,g))) is set
g is V11() real ext-real Element of REAL
(n (#) T) | x0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
(n (#) S) | x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
g is V11() real ext-real Element of REAL
(m,(n (#) S),g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(m,S,g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
n * (m,S,g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
K395(n) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (n,(id REAL)) is set
(m,S,g) (#) K395(n) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m,(n (#) S),x0) . g is Relation-like Function-like complex-valued ext-real-valued real-valued set
g is V11() real ext-real Element of REAL
(m,(n (#) S),x0) . g is Relation-like Function-like complex-valued ext-real-valued real-valued set
(m,S,g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
n * (m,S,g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(m,S,g) (#) K395(n) is Relation-like Function-like complex-valued ext-real-valued real-valued set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is V160() V161() V162() open Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom x0 is V160() V161() V162() Element of K6(REAL)
- x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
(n,(- x0),m) is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
(- 1) (#) x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
K677(REAL,(REAL n),x0,(- 1)) is Relation-like REAL -defined K640(K635((REAL n))) -valued Function-like V239() V240() V241() Element of K6(K7(REAL,K640(K635((REAL n)))))
K635((REAL n)) is set
K640(K635((REAL n))) is functional V233() V234() V235() set
K7(REAL,K640(K635((REAL n)))) is set
K6(K7(REAL,K640(K635((REAL n))))) is set
S is V11() real ext-real Element of REAL
(n,(- x0),m) . S is Relation-like Function-like complex-valued ext-real-valued real-valued set
(n,x0,S) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
- (n,x0,S) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K446() is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
(n,x0,S) (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) (n,x0,S) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K395(K38(1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (K38(1),(id REAL)) is set
(n,x0,S) (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is V160() V161() V162() open Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
S is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 + S is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
K708(REAL,REAL,(REAL n),(REAL n),x0,S) is Relation-like REAL /\ REAL -defined K640((K635((REAL n)) /\ K635((REAL n)))) -valued Function-like V239() V240() V241() Element of K6(K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n))))))
REAL /\ REAL is V160() V161() V162() set
K635((REAL n)) is set
K635((REAL n)) /\ K635((REAL n)) is set
K640((K635((REAL n)) /\ K635((REAL n)))) is functional V233() V234() V235() set
K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n))))) is set
K6(K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n)))))) is set
dom (x0 + S) is V160() V161() V162() Element of K6(REAL)
(n,(x0 + S),m) is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
g is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T + g is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom (T + g) is V160() V161() V162() Element of K6(REAL)
dom T is V160() V161() V162() Element of K6(REAL)
dom g is V160() V161() V162() Element of K6(REAL)
N1 is V11() real ext-real Element of REAL
x0 | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
T | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
N1 is V11() real ext-real Element of REAL
S | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
g | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
(T + g) `| m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
N1 is V11() real ext-real Element of REAL
((T + g) `| m) . N1 is set
diff (T,N1) is Element of the carrier of (REAL-NS n)
diff (g,N1) is Element of the carrier of (REAL-NS n)
(diff (T,N1)) + (diff (g,N1)) is Element of the carrier of (REAL-NS n)
N1 is V11() real ext-real Element of REAL
(T + g) | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
(x0 + S) | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
N1 is V11() real ext-real Element of REAL
(n,(x0 + S),m) . N1 is Relation-like Function-like complex-valued ext-real-valued real-valued set
(n,x0,N1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
(n,S,N1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
(n,x0,N1) + (n,S,N1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() .: ((n,x0,N1),(n,S,N1)) is set
(n,(x0 + S),N1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is V160() V161() V162() open Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
S is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 - S is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
K714(REAL,REAL,(REAL n),(REAL n),x0,S) is Relation-like REAL /\ REAL -defined K640((K635((REAL n)) /\ K635((REAL n)))) -valued Function-like V239() V240() V241() Element of K6(K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n))))))
REAL /\ REAL is V160() V161() V162() set
K635((REAL n)) is set
K635((REAL n)) /\ K635((REAL n)) is set
K640((K635((REAL n)) /\ K635((REAL n)))) is functional V233() V234() V235() set
K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n))))) is set
K6(K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n)))))) is set
dom (x0 - S) is V160() V161() V162() Element of K6(REAL)
(n,(x0 - S),m) is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
- S is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 + (- S) is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
K708(REAL,REAL,(REAL n),(REAL n),x0,(- S)) is Relation-like REAL /\ REAL -defined K640((K635((REAL n)) /\ K635((REAL n)))) -valued Function-like V239() V240() V241() Element of K6(K7((REAL /\ REAL),K640((K635((REAL n)) /\ K635((REAL n))))))
dom x0 is V160() V161() V162() Element of K6(REAL)
dom S is V160() V161() V162() Element of K6(REAL)
(dom x0) /\ (dom S) is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
(n,(x0 - S),m) . T is Relation-like Function-like complex-valued ext-real-valued real-valued set
(n,x0,T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
(n,S,T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
(n,x0,T) - (n,S,T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K449() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K446() is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K448() * ((id REAL),K446()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K449() .: ((n,x0,T),(n,S,T)) is set
- (n,S,T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
(n,S,T) (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) (n,S,T) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K395(K38(1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K450() [;] (K38(1),(id REAL)) is set
(n,S,T) (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(n,x0,T) + (- (n,S,T)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: ((n,x0,T),(- (n,S,T))) is set
(n,(- S),T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
(n,x0,T) + (n,(- S),T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K448() .: ((n,x0,T),(n,(- S),T)) is set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
0* n is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of n -tuples_on REAL
m is V160() V161() V162() open Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom x0 is V160() V161() V162() Element of K6(REAL)
x0 | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
(n,x0,m) is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
S is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
S | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
S `| m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
T is V11() real ext-real Element of REAL
(S `| m) . T is set
T is V11() real ext-real Element of REAL
T is V11() real ext-real Element of REAL
(n,x0,m) . T is Relation-like Function-like complex-valued ext-real-valued real-valued set
(S `| m) . T is set
diff (S,T) is Element of the carrier of (REAL-NS n)
(n,x0,T) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is V160() V161() V162() open Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom x0 is V160() V161() V162() Element of K6(REAL)
(n,x0,m) is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
S is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
N2 is V11() real ext-real Element of REAL
x0 /. N2 is Relation-like Function-like complex-valued ext-real-valued real-valued V56(n) Element of REAL n
N2 * S is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K395(N2) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (N2,(id REAL)) is set
S (#) K395(N2) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(N2 * S) + T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() .: ((N2 * S),T) is set
g is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
g /. N2 is Element of the carrier of (REAL-NS n)
N1 is Element of the carrier of (REAL-NS n)
N2 * N1 is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),N2,N1) is set
f is Element of the carrier of (REAL-NS n)
(N2 * N1) + f is Element of the carrier of (REAL-NS n)
g `| m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
N2 is V11() real ext-real Element of REAL
(g `| m) . N2 is set
N2 is V11() real ext-real Element of REAL
g | m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
x0 | m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
N2 is V11() real ext-real Element of REAL
(n,x0,m) . N2 is Relation-like Function-like complex-valued ext-real-valued real-valued set
(g `| m) . N2 is set
diff (g,N2) is Element of the carrier of (REAL-NS n)
(n,x0,N2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 is V11() real ext-real Element of REAL
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
S is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
n is set
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
x0 | n is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
S is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
dom S is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
S | n is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
n is set
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
x0 is V160() V161() V162() open Element of K6(REAL)
S is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
dom S is V160() V161() V162() Element of K6(REAL)
S | n is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
T is V11() real ext-real Element of REAL
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
g is V11() real ext-real Element of REAL
T is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
T | n is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
g is V11() real ext-real Element of REAL
T | x0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
S | x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
0* n is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of n -tuples_on REAL
REAL --> (0* n) is non empty Relation-like REAL -defined REAL n -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
dom (REAL --> (0* n)) is V160() V161() V162() Element of K6(REAL)
x0 is V11() real ext-real Element of REAL
(REAL --> (0* n)) /. x0 is Relation-like Function-like complex-valued ext-real-valued real-valued V56(n) Element of REAL n
(REAL --> (0* n)) . x0 is Relation-like Function-like complex-valued ext-real-valued real-valued V56(n) Element of REAL n
TOP-REAL n is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like V202() L20()
0. (TOP-REAL n) is real-valued V56(n) FinSequence-like V68( TOP-REAL n) Element of the carrier of (TOP-REAL n)
the carrier of (TOP-REAL n) is non empty set
the ZeroF of (TOP-REAL n) is real-valued V56(n) FinSequence-like Element of the carrier of (TOP-REAL n)
x0 * (0. (TOP-REAL n)) is real-valued V56(n) FinSequence-like Element of the carrier of (TOP-REAL n)
the Mult of (TOP-REAL n) is Relation-like K7(REAL, the carrier of (TOP-REAL n)) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (TOP-REAL n)), the carrier of (TOP-REAL n)))
K7(REAL, the carrier of (TOP-REAL n)) is set
K7(K7(REAL, the carrier of (TOP-REAL n)), the carrier of (TOP-REAL n)) is set
K6(K7(K7(REAL, the carrier of (TOP-REAL n)), the carrier of (TOP-REAL n))) is set
K466( the Mult of (TOP-REAL n),x0,(0. (TOP-REAL n))) is set
(x0 * (0. (TOP-REAL n))) + (0. (TOP-REAL n)) is real-valued V56(n) FinSequence-like Element of the carrier of (TOP-REAL n)
x0 (#) (0* n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like Element of K6(K7(NAT,REAL))
K395(x0) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (x0,(id REAL)) is set
(0* n) (#) K395(x0) is Relation-like Function-like complex-valued ext-real-valued real-valued set
x0 * (0* n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
(x0 * (0* n)) + (0* n) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() .: ((x0 * (0* n)),(0* n)) is set
x0 is non empty Relation-like REAL -defined REAL n -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
0* n is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
n |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of n -tuples_on REAL
REAL --> (0* n) is non empty Relation-like REAL -defined REAL n -valued REAL n -valued Function-like total quasi_total quasi_total V239() V240() V241() (n) Element of K6(K7(REAL,(REAL n)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
m is V160() V161() V162() open Element of K6(REAL)
x0 is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() (n) Element of K6(K7(REAL,(REAL n)))
dom x0 is V160() V161() V162() Element of K6(REAL)
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
x0 is V11() real ext-real Element of REAL
x0 is V11() real ext-real Element of REAL
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
S + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
1 / (S + 1) is V11() real ext-real non negative V159() Element of RAT
K39((S + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((S + 1))) is V11() real ext-real non negative set
S is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
T is V11() real ext-real set
T " is V11() real ext-real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
f is V11() real ext-real set
g + f is V11() real ext-real Element of REAL
N1 is V11() real ext-real set
(T ") + N1 is V11() real ext-real Element of REAL
1 / (T ") is V11() real ext-real Element of REAL
K39((T ")) is V11() real ext-real set
K37(1,K39((T "))) is V11() real ext-real set
g + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
1 / (g + 1) is V11() real ext-real non negative V159() Element of RAT
K39((g + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((g + 1))) is V11() real ext-real non negative set
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
R2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
N2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
R2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
1 / (R2 + 1) is V11() real ext-real non negative V159() Element of RAT
K39((R2 + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((R2 + 1))) is V11() real ext-real non negative set
1 / (N2 + 1) is V11() real ext-real non negative V159() Element of RAT
K39((N2 + 1)) is non empty V11() real ext-real positive non negative set
K37(1,K39((N2 + 1))) is V11() real ext-real non negative set
S . R2 is V11() real ext-real Element of REAL
(S . R2) - 0 is V11() real ext-real Element of REAL
K38(0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() set
K36((S . R2),K38(0)) is V11() real ext-real set
|.((S . R2) - 0).| is V11() real ext-real Element of REAL
lim S is V11() real ext-real Element of REAL
T is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
m /* T is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
K7(NAT, the carrier of (REAL-NS n)) is set
K6(K7(NAT, the carrier of (REAL-NS n))) is set
T " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(T ") (#) (m /* T) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
lim ((T ") (#) (m /* T)) is Element of the carrier of (REAL-NS n)
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
g is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
((T ") (#) (m /* T)) . g is Element of the carrier of (REAL-NS n)
(((T ") (#) (m /* T)) . g) - (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
- (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(((T ") (#) (m /* T)) . g),(- (0. (REAL-NS n)))) is Element of the carrier of (REAL-NS n)
||.((((T ") (#) (m /* T)) . g) - (0. (REAL-NS n))).|| is V11() real ext-real non negative Element of REAL
T . g is V11() real ext-real Element of REAL
m /. (T . g) is Element of the carrier of (REAL-NS n)
(T . g) " is V11() real ext-real Element of REAL
((T . g) ") * (m /. (T . g)) is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),((T . g) "),(m /. (T . g))) is set
||.(((T . g) ") * (m /. (T . g))).|| is V11() real ext-real non negative Element of REAL
abs ((T . g) ") is V11() real ext-real Element of REAL
||.(m /. (T . g)).|| is V11() real ext-real non negative Element of REAL
(abs ((T . g) ")) * ||.(m /. (T . g)).|| is V11() real ext-real Element of REAL
|.(T . g).| is V11() real ext-real Element of REAL
|.(T . g).| " is V11() real ext-real Element of REAL
(|.(T . g).| ") * ||.(m /. (T . g)).|| is V11() real ext-real Element of REAL
dom m is V160() V161() V162() Element of K6(REAL)
rng T is V160() V161() V162() Element of K6(REAL)
||.(((T ") (#) (m /* T)) . g).|| is V11() real ext-real non negative Element of REAL
(m /* T) . g is Element of the carrier of (REAL-NS n)
(T ") . g is V11() real ext-real Element of REAL
((T ") . g) * ((m /* T) . g) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),((T ") . g),((m /* T) . g)) is set
||.(((T ") . g) * ((m /* T) . g)).|| is V11() real ext-real non negative Element of REAL
((T . g) ") * ((m /* T) . g) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),((T . g) "),((m /* T) . g)) is set
||.(((T . g) ") * ((m /* T) . g)).|| is V11() real ext-real non negative Element of REAL
N1 is V11() real ext-real Element of REAL
x0 is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
lim x0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() Element of REAL
S is V11() real ext-real Element of REAL
T is V11() real ext-real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
m /* x0 is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
x0 " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(x0 ") (#) (m /* x0) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
rng x0 is V160() V161() V162() Element of K6(REAL)
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
((x0 ") (#) (m /* x0)) . f is Element of the carrier of (REAL-NS n)
(((x0 ") (#) (m /* x0)) . f) - (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(((x0 ") (#) (m /* x0)) . f),(- (0. (REAL-NS n)))) is Element of the carrier of (REAL-NS n)
||.((((x0 ") (#) (m /* x0)) . f) - (0. (REAL-NS n))).|| is V11() real ext-real non negative Element of REAL
x0 . f is V11() real ext-real Element of REAL
(x0 . f) - 0 is V11() real ext-real Element of REAL
K36((x0 . f),K38(0)) is V11() real ext-real set
|.((x0 . f) - 0).| is V11() real ext-real Element of REAL
|.(x0 . f).| is V11() real ext-real Element of REAL
|.(x0 . f).| " is V11() real ext-real Element of REAL
m /. (x0 . f) is Element of the carrier of (REAL-NS n)
||.(m /. (x0 . f)).|| is V11() real ext-real non negative Element of REAL
(|.(x0 . f).| ") * ||.(m /. (x0 . f)).|| is V11() real ext-real Element of REAL
(x0 . f) " is V11() real ext-real Element of REAL
abs ((x0 . f) ") is V11() real ext-real Element of REAL
(abs ((x0 . f) ")) * ||.(m /. (x0 . f)).|| is V11() real ext-real Element of REAL
((x0 . f) ") * (m /. (x0 . f)) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),((x0 . f) "),(m /. (x0 . f))) is set
||.(((x0 . f) ") * (m /. (x0 . f))).|| is V11() real ext-real non negative Element of REAL
(m /* x0) . f is Element of the carrier of (REAL-NS n)
((x0 . f) ") * ((m /* x0) . f) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),((x0 . f) "),((m /* x0) . f)) is set
||.(((x0 . f) ") * ((m /* x0) . f)).|| is V11() real ext-real non negative Element of REAL
(x0 ") . f is V11() real ext-real Element of REAL
((x0 ") . f) * ((m /* x0) . f) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),((x0 ") . f),((m /* x0) . f)) is set
||.(((x0 ") . f) * ((m /* x0) . f)).|| is V11() real ext-real non negative Element of REAL
||.(((x0 ") (#) (m /* x0)) . f).|| is V11() real ext-real non negative Element of REAL
lim ((x0 ") (#) (m /* x0)) is Element of the carrier of (REAL-NS n)
x0 is V11() real ext-real Element of REAL
S is V11() real ext-real Element of REAL
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
Proj (n,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
x0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
(Proj (n,m)) * x0 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
S is V11() real ext-real set
diff (x0,S) is Element of the carrier of (REAL-NS m)
(Proj (n,m)) . (diff (x0,S)) is Element of the carrier of (REAL-NS 1)
diff (((Proj (n,m)) * x0),S) is Element of the carrier of (REAL-NS 1)
dom x0 is V160() V161() V162() Element of K6(REAL)
x0 /. S is Element of the carrier of (REAL-NS m)
T is V160() V161() V162() open Neighbourhood of S
g is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
g . 1 is Element of the carrier of (REAL-NS m)
N1 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N1 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N1 . 1 is Element of the carrier of (REAL-NS m)
g is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
f is Element of the carrier of (REAL-NS m)
(Proj (n,m)) * N1 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
R2 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
(Proj (n,m)) . f is Element of the carrier of (REAL-NS 1)
L2 is V11() real ext-real Element of REAL
R2 . L2 is Element of the carrier of (REAL-NS 1)
L2 * ((Proj (n,m)) . f) is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),L2,((Proj (n,m)) . f)) is set
dom R2 is V160() V161() V162() Element of K6(REAL)
N1 . L2 is Element of the carrier of (REAL-NS m)
L2 * f is Element of the carrier of (REAL-NS m)
the Mult of (REAL-NS m) is Relation-like K7(REAL, the carrier of (REAL-NS m)) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m)))
K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m))) is set
K466( the Mult of (REAL-NS m),L2,f) is set
(Proj (n,m)) . (N1 . L2) is Element of the carrier of (REAL-NS 1)
L1 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS m)))
(Proj (n,m)) * L1 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
(Proj (n,m)) * g is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
dom g is V160() V161() V162() Element of K6(REAL)
R1 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
r is V11() real ext-real Element of REAL
p0 is V11() real ext-real Element of REAL
q is V11() real ext-real Element of REAL
|.q.| is V11() real ext-real Element of REAL
|.q.| " is V11() real ext-real Element of REAL
R1 /. q is Element of the carrier of (REAL-NS 1)
||.(R1 /. q).|| is V11() real ext-real non negative Element of REAL
(|.q.| ") * ||.(R1 /. q).|| is V11() real ext-real Element of REAL
g /. q is Element of the carrier of (REAL-NS m)
g . q is set
Seg m is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT : ( 1 <= b₁ & b₁ <= m ) } is set
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
L0 is Element of the carrier of (REAL-NS m)
N3 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
N3 . n is V11() real ext-real Element of REAL
dom (Proj (n,m)) is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
dom ((Proj (n,m)) * g) is V160() V161() V162() Element of K6(REAL)
((Proj (n,m)) * g) . q is set
(Proj (n,m)) . (g . q) is set
proj (n,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (n,m)) . N3 is V11() real ext-real Element of REAL
<*((proj (n,m)) . N3)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((proj (n,m)) . N3)] is set
{1,((proj (n,m)) . N3)} is non empty V160() V161() V162() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,((proj (n,m)) . N3)},{1}} is non empty set
{[1,((proj (n,m)) . N3)]} is non empty Function-like set
R0 is V11() real ext-real Element of REAL
<*R0*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,R0] is set
{1,R0} is non empty V160() V161() V162() set
{{1,R0},{1}} is non empty set
{[1,R0]} is non empty Function-like set
abs R0 is V11() real ext-real Element of REAL
||.(g /. q).|| is V11() real ext-real non negative Element of REAL
(|.q.| ") * (abs R0) is V11() real ext-real Element of REAL
(|.q.| ") * ||.(g /. q).|| is V11() real ext-real Element of REAL
rng ((Proj (n,m)) * g) is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
N is Element of the carrier of (REAL-NS 1)
||.N.|| is V11() real ext-real non negative Element of REAL
|.<*R0*>.| is V11() real ext-real non negative Element of REAL
sqr <*R0*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
<*R0*> (#) sqrreal is Relation-like Function-like complex-valued ext-real-valued real-valued set
<*R0*> (#) <*R0*> is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K450() .: (<*R0*>,<*R0*>) is set
Sum (sqr <*R0*>) is V11() real ext-real Element of REAL
sqrt (Sum (sqr <*R0*>)) is V11() real ext-real Element of REAL
(|.q.| ") * ||.N.|| is V11() real ext-real Element of REAL
dom (Proj (n,m)) is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
rng x0 is Element of K6( the carrier of (REAL-NS m))
dom ((Proj (n,m)) * x0) is V160() V161() V162() Element of K6(REAL)
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
((Proj (n,m)) * x0) /. S is Element of the carrier of (REAL-NS 1)
L2 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
R1 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
p0 is V11() real ext-real Element of REAL
((Proj (n,m)) * x0) /. p0 is Element of the carrier of (REAL-NS 1)
(((Proj (n,m)) * x0) /. p0) - (((Proj (n,m)) * x0) /. S) is Element of the carrier of (REAL-NS 1)
- (((Proj (n,m)) * x0) /. S) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(((Proj (n,m)) * x0) /. p0),(- (((Proj (n,m)) * x0) /. S))) is Element of the carrier of (REAL-NS 1)
p0 - S is V11() real ext-real Element of REAL
K38(S) is V11() real ext-real set
K36(p0,K38(S)) is V11() real ext-real set
L2 . (p0 - S) is Element of the carrier of (REAL-NS 1)
R1 /. (p0 - S) is Element of the carrier of (REAL-NS 1)
(L2 . (p0 - S)) + (R1 /. (p0 - S)) is Element of the carrier of (REAL-NS 1)
x0 /. p0 is Element of the carrier of (REAL-NS m)
(x0 /. p0) - (x0 /. S) is Element of the carrier of (REAL-NS m)
- (x0 /. S) is Element of the carrier of (REAL-NS m)
K192((REAL-NS m),(x0 /. p0),(- (x0 /. S))) is Element of the carrier of (REAL-NS m)
N1 . (p0 - S) is Element of the carrier of (REAL-NS m)
g /. (p0 - S) is Element of the carrier of (REAL-NS m)
(N1 . (p0 - S)) + (g /. (p0 - S)) is Element of the carrier of (REAL-NS m)
((Proj (n,m)) * x0) . p0 is set
((Proj (n,m)) * x0) . S is set
q is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
q . 1 is Element of the carrier of (REAL-NS m)
L0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
x0 . p0 is set
x0 . S is set
q is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
q . 1 is Element of the carrier of (REAL-NS m)
L0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
rng ((Proj (n,m)) * x0) is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
L0 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
L0 . 1 is Element of the carrier of (REAL-NS m)
N3 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N3 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N3 . 1 is Element of the carrier of (REAL-NS m)
R0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
R0 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
R0 . 1 is Element of the carrier of (REAL-NS m)
N is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N . 1 is Element of the carrier of (REAL-NS m)
x is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
(Proj (n,m)) . (x0 . p0) is set
R0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
rng (Proj (n,m)) is Element of K6( the carrier of (REAL-NS 1))
(Proj (n,m)) . (x0 . S) is set
N is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
dom R1 is V160() V161() V162() Element of K6(REAL)
R1 . (p0 - S) is set
Ldiff is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(Proj (n,m)) . Ldiff is set
dom g is V160() V161() V162() Element of K6(REAL)
g . (p0 - S) is set
rng g is Element of K6( the carrier of (REAL-NS m))
Rdiff is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(Proj (n,m)) . Rdiff is set
dom L2 is V160() V161() V162() Element of K6(REAL)
Ldxx0 is Element of the carrier of (REAL-NS 1)
Rdxx0 is Element of the carrier of (REAL-NS 1)
Ldxx0 + Rdxx0 is Element of the carrier of (REAL-NS 1)
ProjLdiff is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
ProjRdiff is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
ProjLdiff + ProjRdiff is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() .: (ProjLdiff,ProjRdiff) is set
proj (n,m) is non empty Relation-like REAL m -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL m),REAL))
K7((REAL m),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL m),REAL)) is set
(proj (n,m)) . Ldiff is V11() real ext-real Element of REAL
<*((proj (n,m)) . Ldiff)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((proj (n,m)) . Ldiff)] is set
{1,((proj (n,m)) . Ldiff)} is non empty V160() V161() V162() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,((proj (n,m)) . Ldiff)},{1}} is non empty set
{[1,((proj (n,m)) . Ldiff)]} is non empty Function-like set
Ldiff . n is V11() real ext-real Element of REAL
<*(Ldiff . n)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,(Ldiff . n)] is set
{1,(Ldiff . n)} is non empty V160() V161() V162() set
{{1,(Ldiff . n)},{1}} is non empty set
{[1,(Ldiff . n)]} is non empty Function-like set
(proj (n,m)) . Rdiff is V11() real ext-real Element of REAL
<*((proj (n,m)) . Rdiff)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((proj (n,m)) . Rdiff)] is set
{1,((proj (n,m)) . Rdiff)} is non empty V160() V161() V162() set
{{1,((proj (n,m)) . Rdiff)},{1}} is non empty set
{[1,((proj (n,m)) . Rdiff)]} is non empty Function-like set
Rdiff . n is V11() real ext-real Element of REAL
<*(Rdiff . n)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,(Rdiff . n)] is set
{1,(Rdiff . n)} is non empty V160() V161() V162() set
{{1,(Rdiff . n)},{1}} is non empty set
{[1,(Rdiff . n)]} is non empty Function-like set
q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
L0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
N3 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
L0 - N3 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K449() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K446() is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K448() * ((id REAL),K446()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K449() .: (L0,N3) is set
- N3 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
N3 (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) N3 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K395(K38(1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K450() [;] (K38(1),(id REAL)) is set
N3 (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
L0 + (- N3) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (L0,(- N3)) is set
x9 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
x9 - N3 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K449() .: (x9,N3) is set
x9 + (- N3) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (x9,(- N3)) is set
ProjGx0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
x9 - ProjGx0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K449() .: (x9,ProjGx0) is set
- ProjGx0 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
ProjGx0 (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) (#) ProjGx0 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
ProjGx0 (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
x9 + (- ProjGx0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (x9,(- ProjGx0)) is set
Gx1 is Element of the carrier of (REAL-NS m)
(proj (n,m)) . Gx1 is V11() real ext-real Element of REAL
<*((proj (n,m)) . Gx1)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((proj (n,m)) . Gx1)] is set
{1,((proj (n,m)) . Gx1)} is non empty V160() V161() V162() set
{{1,((proj (n,m)) . Gx1)},{1}} is non empty set
{[1,((proj (n,m)) . Gx1)]} is non empty Function-like set
<*((proj (n,m)) . Gx1)*> - ProjGx0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
K449() .: (<*((proj (n,m)) . Gx1)*>,ProjGx0) is set
<*((proj (n,m)) . Gx1)*> + (- ProjGx0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (<*((proj (n,m)) . Gx1)*>,(- ProjGx0)) is set
Gx01 is Element of the carrier of (REAL-NS m)
(proj (n,m)) . Gx01 is V11() real ext-real Element of REAL
<*((proj (n,m)) . Gx01)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((proj (n,m)) . Gx01)] is set
{1,((proj (n,m)) . Gx01)} is non empty V160() V161() V162() set
{{1,((proj (n,m)) . Gx01)},{1}} is non empty set
{[1,((proj (n,m)) . Gx01)]} is non empty Function-like set
<*((proj (n,m)) . Gx1)*> - <*((proj (n,m)) . Gx01)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
K449() .: (<*((proj (n,m)) . Gx1)*>,<*((proj (n,m)) . Gx01)*>) is set
- <*((proj (n,m)) . Gx01)*> is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
<*((proj (n,m)) . Gx01)*> (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) (#) <*((proj (n,m)) . Gx01)*> is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
<*((proj (n,m)) . Gx01)*> (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
<*((proj (n,m)) . Gx1)*> + (- <*((proj (n,m)) . Gx01)*>) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (<*((proj (n,m)) . Gx1)*>,(- <*((proj (n,m)) . Gx01)*>)) is set
R0 . n is V11() real ext-real Element of REAL
<*(R0 . n)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,(R0 . n)] is set
{1,(R0 . n)} is non empty V160() V161() V162() set
{{1,(R0 . n)},{1}} is non empty set
{[1,(R0 . n)]} is non empty Function-like set
<*(R0 . n)*> - <*((proj (n,m)) . Gx01)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
K449() .: (<*(R0 . n)*>,<*((proj (n,m)) . Gx01)*>) is set
<*(R0 . n)*> + (- <*((proj (n,m)) . Gx01)*>) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (<*(R0 . n)*>,(- <*((proj (n,m)) . Gx01)*>)) is set
N . n is V11() real ext-real Element of REAL
<*(N . n)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,(N . n)] is set
{1,(N . n)} is non empty V160() V161() V162() set
{{1,(N . n)},{1}} is non empty set
{[1,(N . n)]} is non empty Function-like set
<*(R0 . n)*> - <*(N . n)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
K449() .: (<*(R0 . n)*>,<*(N . n)*>) is set
- <*(N . n)*> is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
<*(N . n)*> (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) (#) <*(N . n)*> is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
<*(N . n)*> (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
<*(R0 . n)*> + (- <*(N . n)*>) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (<*(R0 . n)*>,(- <*(N . n)*>)) is set
(R0 . n) - (N . n) is V11() real ext-real Element of REAL
K38((N . n)) is V11() real ext-real set
K36((R0 . n),K38((N . n))) is V11() real ext-real set
<*((R0 . n) - (N . n))*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((R0 . n) - (N . n))] is set
{1,((R0 . n) - (N . n))} is non empty V160() V161() V162() set
{{1,((R0 . n) - (N . n))},{1}} is non empty set
{[1,((R0 . n) - (N . n))]} is non empty Function-like set
Gsx is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
Gsx0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
Gsx - Gsx0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
K449() .: (Gsx,Gsx0) is set
- Gsx0 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
Gsx0 (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) (#) Gsx0 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
Gsx0 (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
Gsx + (- Gsx0) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (Gsx,(- Gsx0)) is set
(Gsx - Gsx0) . n is V11() real ext-real Element of REAL
<*((Gsx - Gsx0) . n)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((Gsx - Gsx0) . n)] is set
{1,((Gsx - Gsx0) . n)} is non empty V160() V161() V162() set
{{1,((Gsx - Gsx0) . n)},{1}} is non empty set
{[1,((Gsx - Gsx0) . n)]} is non empty Function-like set
diffGR is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
diffGR . n is V11() real ext-real Element of REAL
<*(diffGR . n)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,(diffGR . n)] is set
{1,(diffGR . n)} is non empty V160() V161() V162() set
{{1,(diffGR . n)},{1}} is non empty set
{[1,(diffGR . n)]} is non empty Function-like set
Rsdiff is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
Ldiff + Rsdiff is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
K448() .: (Ldiff,Rsdiff) is set
(Ldiff + Rsdiff) . n is V11() real ext-real Element of REAL
<*((Ldiff + Rsdiff) . n)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((Ldiff + Rsdiff) . n)] is set
{1,((Ldiff + Rsdiff) . n)} is non empty V160() V161() V162() set
{{1,((Ldiff + Rsdiff) . n)},{1}} is non empty set
{[1,((Ldiff + Rsdiff) . n)]} is non empty Function-like set
Rsdiff . n is V11() real ext-real Element of REAL
(Ldiff . n) + (Rsdiff . n) is V11() real ext-real Element of REAL
<*((Ldiff . n) + (Rsdiff . n))*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((Ldiff . n) + (Rsdiff . n))] is set
{1,((Ldiff . n) + (Rsdiff . n))} is non empty V160() V161() V162() set
{{1,((Ldiff . n) + (Rsdiff . n))},{1}} is non empty set
{[1,((Ldiff . n) + (Rsdiff . n))]} is non empty Function-like set
c₄₃ is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
c₄₃ . 1 is Element of the carrier of (REAL-NS m)
c₄₄ is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
c₄₃ is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
c₄₃ . 1 is Element of the carrier of (REAL-NS m)
c₄₄ is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
p0 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
p0 . 1 is Element of the carrier of (REAL-NS m)
q is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
L2 . 1 is Element of the carrier of (REAL-NS 1)
1 * ((Proj (n,m)) . f) is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),1,((Proj (n,m)) . f)) is set
1 * f is Element of the carrier of (REAL-NS m)
the Mult of (REAL-NS m) is Relation-like K7(REAL, the carrier of (REAL-NS m)) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m)))
K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m))) is set
K466( the Mult of (REAL-NS m),1,f) is set
(Proj (n,m)) . (1 * f) is Element of the carrier of (REAL-NS 1)
p0 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
p0 . 1 is Element of the carrier of (REAL-NS m)
q is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
x0 is V11() real ext-real set
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (S,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (S,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
Seg n is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT : ( 1 <= b₁ & b₁ <= n ) } is set
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (S,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (S,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
dom ((Proj (S,n)) * m) is V160() V161() V162() Element of K6(REAL)
((Proj (S,n)) * m) /. x0 is Element of the carrier of (REAL-NS 1)
diff (((Proj (S,n)) * m),x0) is Element of the carrier of (REAL-NS 1)
T is V160() V161() V162() open Neighbourhood of x0
g is V11() real ext-real set
x0 - g is V11() real ext-real Element of REAL
K38(g) is V11() real ext-real set
K36(x0,K38(g)) is V11() real ext-real set
x0 + g is V11() real ext-real Element of REAL
].(x0 - g),(x0 + g).[ is V160() V161() V162() open Element of K6(REAL)
N1 is V11() real ext-real Element of REAL
x0 - N1 is V11() real ext-real Element of REAL
K38(N1) is V11() real ext-real set
K36(x0,K38(N1)) is V11() real ext-real set
x0 + N1 is V11() real ext-real Element of REAL
].(x0 - N1),(x0 + N1).[ is V160() V161() V162() open Element of K6(REAL)
f is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
f . 1 is Element of the carrier of (REAL-NS 1)
N2 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
f is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
f . 1 is Element of the carrier of (REAL-NS 1)
N2 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
f is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
f . 1 is Element of the carrier of (REAL-NS 1)
N2 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
R2 is V11() real ext-real Element of REAL
L2 is Element of the carrier of (REAL-NS 1)
1 * L2 is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),1,L2) is set
R2 - x0 is V11() real ext-real Element of REAL
K38(x0) is V11() real ext-real set
K36(R2,K38(x0)) is V11() real ext-real set
f . (R2 - x0) is Element of the carrier of (REAL-NS 1)
(R2 - x0) * (diff (((Proj (S,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),(R2 - x0),(diff (((Proj (S,n)) * m),x0))) is set
((Proj (S,n)) * m) /. R2 is Element of the carrier of (REAL-NS 1)
(((Proj (S,n)) * m) /. R2) - (((Proj (S,n)) * m) /. x0) is Element of the carrier of (REAL-NS 1)
- (((Proj (S,n)) * m) /. x0) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(((Proj (S,n)) * m) /. R2),(- (((Proj (S,n)) * m) /. x0))) is Element of the carrier of (REAL-NS 1)
N2 /. (R2 - x0) is Element of the carrier of (REAL-NS 1)
((R2 - x0) * (diff (((Proj (S,n)) * m),x0))) + (N2 /. (R2 - x0)) is Element of the carrier of (REAL-NS 1)
R2 is V11() real ext-real Element of REAL
((Proj (S,n)) * m) /. R2 is Element of the carrier of (REAL-NS 1)
(((Proj (S,n)) * m) /. R2) - (((Proj (S,n)) * m) /. x0) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(((Proj (S,n)) * m) /. R2),(- (((Proj (S,n)) * m) /. x0))) is Element of the carrier of (REAL-NS 1)
R2 - x0 is V11() real ext-real Element of REAL
K36(R2,K38(x0)) is V11() real ext-real set
(R2 - x0) * (diff (((Proj (S,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),(R2 - x0),(diff (((Proj (S,n)) * m),x0))) is set
N2 /. (R2 - x0) is Element of the carrier of (REAL-NS 1)
((R2 - x0) * (diff (((Proj (S,n)) * m),x0))) + (N2 /. (R2 - x0)) is Element of the carrier of (REAL-NS 1)
S is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
dom S is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
min S is V11() real ext-real Element of REAL
min_p S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
S . (min_p S) is V11() real ext-real Element of REAL
len S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
S /. (min_p S) is V11() real ext-real Element of REAL
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
g is V11() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (N1,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (N1,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
diff (((Proj (N1,n)) * m),x0) is Element of the carrier of (REAL-NS 1)
g * (diff (((Proj (N1,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),g,(diff (((Proj (N1,n)) * m),x0))) is set
(proj (1,1)) . (g * (diff (((Proj (N1,n)) * m),x0))) is V11() real ext-real Element of REAL
N1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
dom N1 is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
len N1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
f /. N2 is V11() real ext-real Element of REAL
Proj (N2,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (N2,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
diff (((Proj (N2,n)) * m),x0) is Element of the carrier of (REAL-NS 1)
g * (diff (((Proj (N2,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),g,(diff (((Proj (N2,n)) * m),x0))) is set
(proj (1,1)) . (g * (diff (((Proj (N2,n)) * m),x0))) is V11() real ext-real Element of REAL
f . N2 is V11() real ext-real Element of REAL
N2 is Element of the carrier of (REAL-NS n)
R2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
g is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS n)))
g . 1 is Element of the carrier of (REAL-NS n)
N1 is V11() real ext-real Element of REAL
g . N1 is Element of the carrier of (REAL-NS n)
N1 * (g . 1) is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),N1,(g . 1)) is set
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
f . N2 is V11() real ext-real Element of REAL
Proj (N2,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (N2,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
diff (((Proj (N2,n)) * m),x0) is Element of the carrier of (REAL-NS 1)
1 * (diff (((Proj (N2,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),1,(diff (((Proj (N2,n)) * m),x0))) is set
(proj (1,1)) . (1 * (diff (((Proj (N2,n)) * m),x0))) is V11() real ext-real Element of REAL
(proj (1,1)) . (diff (((Proj (N2,n)) * m),x0)) is V11() real ext-real Element of REAL
N2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
N1 * f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K395(N1) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (N1,(id REAL)) is set
f (#) K395(N1) is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom (N1 * f) is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
dom N2 is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
R2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(N1 * f) . R2 is V11() real ext-real Element of REAL
N2 . R2 is V11() real ext-real Element of REAL
L2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
f . L2 is V11() real ext-real Element of REAL
Proj (L2,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
(Proj (L2,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
diff (((Proj (L2,n)) * m),x0) is Element of the carrier of (REAL-NS 1)
(proj (1,1)) . (diff (((Proj (L2,n)) * m),x0)) is V11() real ext-real Element of REAL
N2 . L2 is V11() real ext-real Element of REAL
N1 * (diff (((Proj (L2,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),N1,(diff (((Proj (L2,n)) * m),x0))) is set
(proj (1,1)) . (N1 * (diff (((Proj (L2,n)) * m),x0))) is V11() real ext-real Element of REAL
N1 * 1 is V11() real ext-real Element of REAL
(N1 * 1) * (diff (((Proj (L2,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),(N1 * 1),(diff (((Proj (L2,n)) * m),x0))) is set
1 * (diff (((Proj (L2,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),1,(diff (((Proj (L2,n)) * m),x0))) is set
N1 * (1 * (diff (((Proj (L2,n)) * m),x0))) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),N1,(1 * (diff (((Proj (L2,n)) * m),x0)))) is set
R1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
L1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
N1 * R1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
R1 (#) K395(N1) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(N1 * R1) . 1 is V11() real ext-real Element of REAL
R1 . 1 is V11() real ext-real Element of REAL
N1 * (R1 . 1) is V11() real ext-real Element of REAL
f . R2 is V11() real ext-real Element of REAL
N1 * (f . R2) is V11() real ext-real Element of REAL
N1 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
m /. x0 is Element of the carrier of (REAL-NS n)
N2 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom m is V160() V161() V162() Element of K6(REAL)
R2 is V11() real ext-real Element of REAL
R2 + x0 is V11() real ext-real Element of REAL
L2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (L2,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (L2,n)) * N2 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
((Proj (L2,n)) * N2) . R2 is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
((Proj (L2,n)) * m) /. (R2 + x0) is Element of the carrier of (REAL-NS 1)
((Proj (L2,n)) * m) /. x0 is Element of the carrier of (REAL-NS 1)
(((Proj (L2,n)) * m) /. (R2 + x0)) - (((Proj (L2,n)) * m) /. x0) is Element of the carrier of (REAL-NS 1)
- (((Proj (L2,n)) * m) /. x0) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(((Proj (L2,n)) * m) /. (R2 + x0)),(- (((Proj (L2,n)) * m) /. x0))) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) * N1 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
((Proj (L2,n)) * N1) . R2 is Element of the carrier of (REAL-NS 1)
((((Proj (L2,n)) * m) /. (R2 + x0)) - (((Proj (L2,n)) * m) /. x0)) - (((Proj (L2,n)) * N1) . R2) is Element of the carrier of (REAL-NS 1)
- (((Proj (L2,n)) * N1) . R2) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),((((Proj (L2,n)) * m) /. (R2 + x0)) - (((Proj (L2,n)) * m) /. x0)),(- (((Proj (L2,n)) * N1) . R2))) is Element of the carrier of (REAL-NS 1)
x0 - x0 is V11() real ext-real Element of REAL
K38(x0) is V11() real ext-real set
K36(x0,K38(x0)) is V11() real ext-real set
|.(x0 - x0).| is V11() real ext-real Element of REAL
S /. L2 is V11() real ext-real Element of REAL
x0 - (S /. L2) is V11() real ext-real Element of REAL
K38((S /. L2)) is V11() real ext-real set
K36(x0,K38((S /. L2))) is V11() real ext-real set
x0 + (S /. L2) is V11() real ext-real Element of REAL
].(x0 - (S /. L2)),(x0 + (S /. L2)).[ is V160() V161() V162() open Element of K6(REAL)
dom ((Proj (L2,n)) * m) is V160() V161() V162() Element of K6(REAL)
m /. (R2 + x0) is Element of the carrier of (REAL-NS n)
N1 . R2 is Element of the carrier of (REAL-NS n)
R1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
- R1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K446() is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
R1 (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) R1 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K395(K38(1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (K38(1),(id REAL)) is set
R1 (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
- 1 is V11() real ext-real non positive V132() V159() Element of INT
(- 1) * (m /. x0) is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),(- 1),(m /. x0)) is set
r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
- r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
r (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) (#) r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
r (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(- 1) * (N1 . R2) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),(- 1),(N1 . R2)) is set
L1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
L1 + (- R1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() .: (L1,(- R1)) is set
(m /. (R2 + x0)) + ((- 1) * (m /. x0)) is Element of the carrier of (REAL-NS n)
(L1 + (- R1)) + (- r) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K448() .: ((L1 + (- R1)),(- r)) is set
((m /. (R2 + x0)) + ((- 1) * (m /. x0))) + ((- 1) * (N1 . R2)) is Element of the carrier of (REAL-NS n)
L1 - R1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K449() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() * ((id REAL),K446()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K449() .: (L1,R1) is set
- R1 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
L1 + (- R1) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (L1,(- R1)) is set
(m /. (R2 + x0)) - (m /. x0) is Element of the carrier of (REAL-NS n)
- (m /. x0) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(m /. (R2 + x0)),(- (m /. x0))) is Element of the carrier of (REAL-NS n)
((m /. (R2 + x0)) - (m /. x0)) - (N1 . R2) is Element of the carrier of (REAL-NS n)
- (N1 . R2) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),((m /. (R2 + x0)) - (m /. x0)),(- (N1 . R2))) is Element of the carrier of (REAL-NS n)
(L1 - R1) - r is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
K449() .: ((L1 - R1),r) is set
- r is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
(L1 - R1) + (- r) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: ((L1 - R1),(- r)) is set
N2 . R2 is Element of the carrier of (REAL-NS n)
(Proj (L2,n)) . (N2 . R2) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) . (((m /. (R2 + x0)) - (m /. x0)) - (N1 . R2)) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) . ((m /. (R2 + x0)) + ((- 1) * (m /. x0))) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) . ((- 1) * (N1 . R2)) is Element of the carrier of (REAL-NS 1)
((Proj (L2,n)) . ((m /. (R2 + x0)) + ((- 1) * (m /. x0)))) + ((Proj (L2,n)) . ((- 1) * (N1 . R2))) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) . (m /. (R2 + x0)) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) . ((- 1) * (m /. x0)) is Element of the carrier of (REAL-NS 1)
((Proj (L2,n)) . (m /. (R2 + x0))) + ((Proj (L2,n)) . ((- 1) * (m /. x0))) is Element of the carrier of (REAL-NS 1)
(((Proj (L2,n)) . (m /. (R2 + x0))) + ((Proj (L2,n)) . ((- 1) * (m /. x0)))) + ((Proj (L2,n)) . ((- 1) * (N1 . R2))) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) . (m /. x0) is Element of the carrier of (REAL-NS 1)
(- 1) * ((Proj (L2,n)) . (m /. x0)) is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),(- 1),((Proj (L2,n)) . (m /. x0))) is set
L0 is Element of the carrier of (REAL-NS n)
(- 1) * L0 is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),(- 1),L0) is set
(Proj (L2,n)) . ((- 1) * L0) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) . L0 is Element of the carrier of (REAL-NS 1)
(- 1) * ((Proj (L2,n)) . L0) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),(- 1),((Proj (L2,n)) . L0)) is set
- ((Proj (L2,n)) . (m /. x0)) is Element of the carrier of (REAL-NS 1)
((Proj (L2,n)) . (m /. (R2 + x0))) + (- ((Proj (L2,n)) . (m /. x0))) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) . (N1 . R2) is Element of the carrier of (REAL-NS 1)
(- 1) * ((Proj (L2,n)) . (N1 . R2)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),(- 1),((Proj (L2,n)) . (N1 . R2))) is set
(((Proj (L2,n)) . (m /. (R2 + x0))) + (- ((Proj (L2,n)) . (m /. x0)))) + ((- 1) * ((Proj (L2,n)) . (N1 . R2))) is Element of the carrier of (REAL-NS 1)
((Proj (L2,n)) . (m /. (R2 + x0))) - ((Proj (L2,n)) . (m /. x0)) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),((Proj (L2,n)) . (m /. (R2 + x0))),(- ((Proj (L2,n)) . (m /. x0)))) is Element of the carrier of (REAL-NS 1)
(((Proj (L2,n)) . (m /. (R2 + x0))) - ((Proj (L2,n)) . (m /. x0))) - ((Proj (L2,n)) . (N1 . R2)) is Element of the carrier of (REAL-NS 1)
- ((Proj (L2,n)) . (N1 . R2)) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(((Proj (L2,n)) . (m /. (R2 + x0))) - ((Proj (L2,n)) . (m /. x0))),(- ((Proj (L2,n)) . (N1 . R2)))) is Element of the carrier of (REAL-NS 1)
dom (Proj (L2,n)) is Element of K6( the carrier of (REAL-NS n))
K6( the carrier of (REAL-NS n)) is set
(Proj (L2,n)) /. (m /. (R2 + x0)) is Element of the carrier of (REAL-NS 1)
(Proj (L2,n)) /. (m /. x0) is Element of the carrier of (REAL-NS 1)
R2 is V11() real ext-real Element of REAL
sqrt n is V11() real ext-real Element of REAL
(sqrt n) " is V11() real ext-real Element of REAL
((sqrt n) ") * R2 is V11() real ext-real Element of REAL
L1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (L1,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (L1,n)) * N2 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
(Proj (L1,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
dom ((Proj (L1,n)) * m) is V160() V161() V162() Element of K6(REAL)
diff (((Proj (L1,n)) * m),x0) is Element of the carrier of (REAL-NS 1)
((Proj (L1,n)) * m) /. x0 is Element of the carrier of (REAL-NS 1)
R1 is V160() V161() V162() open Neighbourhood of x0
r is V11() real ext-real set
x0 - r is V11() real ext-real Element of REAL
K38(r) is V11() real ext-real set
K36(x0,K38(r)) is V11() real ext-real set
x0 + r is V11() real ext-real Element of REAL
].(x0 - r),(x0 + r).[ is V160() V161() V162() open Element of K6(REAL)
q is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
q . 1 is Element of the carrier of (REAL-NS 1)
L0 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
q is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
q . 1 is Element of the carrier of (REAL-NS 1)
L0 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
N3 is V11() real ext-real Element of REAL
R0 is Element of the carrier of (REAL-NS 1)
1 * R0 is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),1,R0) is set
N3 - x0 is V11() real ext-real Element of REAL
K38(x0) is V11() real ext-real set
K36(N3,K38(x0)) is V11() real ext-real set
q . (N3 - x0) is Element of the carrier of (REAL-NS 1)
(N3 - x0) * (diff (((Proj (L1,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),(N3 - x0),(diff (((Proj (L1,n)) * m),x0))) is set
((Proj (L1,n)) * m) /. N3 is Element of the carrier of (REAL-NS 1)
(((Proj (L1,n)) * m) /. N3) - (((Proj (L1,n)) * m) /. x0) is Element of the carrier of (REAL-NS 1)
- (((Proj (L1,n)) * m) /. x0) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(((Proj (L1,n)) * m) /. N3),(- (((Proj (L1,n)) * m) /. x0))) is Element of the carrier of (REAL-NS 1)
L0 /. (N3 - x0) is Element of the carrier of (REAL-NS 1)
((N3 - x0) * (diff (((Proj (L1,n)) * m),x0))) + (L0 /. (N3 - x0)) is Element of the carrier of (REAL-NS 1)
N3 is V11() real ext-real Element of REAL
p0 is V11() real ext-real Element of REAL
min (p0,N3) is V11() real ext-real Element of REAL
N is V11() real ext-real Element of REAL
|.N.| is V11() real ext-real Element of REAL
|.N.| " is V11() real ext-real Element of REAL
((Proj (L1,n)) * N2) /. N is Element of the carrier of (REAL-NS 1)
((Proj (L1,n)) * N2) . N is Element of the carrier of (REAL-NS 1)
||.(((Proj (L1,n)) * N2) /. N).|| is V11() real ext-real non negative Element of REAL
(|.N.| ") * ||.(((Proj (L1,n)) * N2) /. N).|| is V11() real ext-real Element of REAL
L0 /. N is Element of the carrier of (REAL-NS 1)
||.(L0 /. N).|| is V11() real ext-real non negative Element of REAL
(|.N.| ") * ||.(L0 /. N).|| is V11() real ext-real Element of REAL
N + x0 is V11() real ext-real Element of REAL
(N + x0) - x0 is V11() real ext-real Element of REAL
K36((N + x0),K38(x0)) is V11() real ext-real set
x0 - p0 is V11() real ext-real Element of REAL
K38(p0) is V11() real ext-real set
K36(x0,K38(p0)) is V11() real ext-real set
x0 + p0 is V11() real ext-real Element of REAL
].(x0 - p0),(x0 + p0).[ is V160() V161() V162() open Element of K6(REAL)
x is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
x . 1 is Element of the carrier of (REAL-NS 1)
x9 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
((Proj (L1,n)) * m) /. (N + x0) is Element of the carrier of (REAL-NS 1)
(((Proj (L1,n)) * m) /. (N + x0)) - (((Proj (L1,n)) * m) /. x0) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(((Proj (L1,n)) * m) /. (N + x0)),(- (((Proj (L1,n)) * m) /. x0))) is Element of the carrier of (REAL-NS 1)
((N + x0) - x0) * (diff (((Proj (L1,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),((N + x0) - x0),(diff (((Proj (L1,n)) * m),x0))) is set
L0 /. ((N + x0) - x0) is Element of the carrier of (REAL-NS 1)
(((N + x0) - x0) * (diff (((Proj (L1,n)) * m),x0))) + (L0 /. ((N + x0) - x0)) is Element of the carrier of (REAL-NS 1)
N * (diff (((Proj (L1,n)) * m),x0)) is Element of the carrier of (REAL-NS 1)
K466( the Mult of (REAL-NS 1),N,(diff (((Proj (L1,n)) * m),x0))) is set
((((Proj (L1,n)) * m) /. (N + x0)) - (((Proj (L1,n)) * m) /. x0)) - (N * (diff (((Proj (L1,n)) * m),x0))) is Element of the carrier of (REAL-NS 1)
- (N * (diff (((Proj (L1,n)) * m),x0))) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),((((Proj (L1,n)) * m) /. (N + x0)) - (((Proj (L1,n)) * m) /. x0)),(- (N * (diff (((Proj (L1,n)) * m),x0))))) is Element of the carrier of (REAL-NS 1)
(Proj (L1,n)) * N1 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
((Proj (L1,n)) * N1) . N is Element of the carrier of (REAL-NS 1)
((((Proj (L1,n)) * m) /. (N + x0)) - (((Proj (L1,n)) * m) /. x0)) - (((Proj (L1,n)) * N1) . N) is Element of the carrier of (REAL-NS 1)
- (((Proj (L1,n)) * N1) . N) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),((((Proj (L1,n)) * m) /. (N + x0)) - (((Proj (L1,n)) * m) /. x0)),(- (((Proj (L1,n)) * N1) . N))) is Element of the carrier of (REAL-NS 1)
N1 . N is Element of the carrier of (REAL-NS n)
x is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
x . L1 is V11() real ext-real Element of REAL
(proj (1,1)) . (N * (diff (((Proj (L1,n)) * m),x0))) is V11() real ext-real Element of REAL
x9 is V11() real ext-real Element of REAL
<*x9*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,x9] is set
{1,x9} is non empty V160() V161() V162() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,x9},{1}} is non empty set
{[1,x9]} is non empty Function-like set
dom N1 is V160() V161() V162() Element of K6(REAL)
(Proj (L1,n)) . (N1 . N) is Element of the carrier of (REAL-NS 1)
proj (L1,n) is non empty Relation-like REAL n -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
K7((REAL n),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL n),REAL)) is set
y1 is Element of the carrier of (REAL-NS n)
(proj (L1,n)) . y1 is V11() real ext-real Element of REAL
<*((proj (L1,n)) . y1)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((proj (L1,n)) . y1)] is set
{1,((proj (L1,n)) . y1)} is non empty V160() V161() V162() set
{{1,((proj (L1,n)) . y1)},{1}} is non empty set
{[1,((proj (L1,n)) . y1)]} is non empty Function-like set
<*((proj (1,1)) . (N * (diff (((Proj (L1,n)) * m),x0))))*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((proj (1,1)) . (N * (diff (((Proj (L1,n)) * m),x0))))] is set
{1,((proj (1,1)) . (N * (diff (((Proj (L1,n)) * m),x0))))} is non empty V160() V161() V162() set
{{1,((proj (1,1)) . (N * (diff (((Proj (L1,n)) * m),x0))))},{1}} is non empty set
{[1,((proj (1,1)) . (N * (diff (((Proj (L1,n)) * m),x0))))]} is non empty Function-like set
N is V11() real ext-real Element of REAL
|.N.| is V11() real ext-real Element of REAL
|.N.| " is V11() real ext-real Element of REAL
((Proj (L1,n)) * N2) /. N is Element of the carrier of (REAL-NS 1)
((Proj (L1,n)) * N2) . N is Element of the carrier of (REAL-NS 1)
||.(((Proj (L1,n)) * N2) /. N).|| is V11() real ext-real non negative Element of REAL
(|.N.| ") * ||.(((Proj (L1,n)) * N2) /. N).|| is V11() real ext-real Element of REAL
L1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
dom L1 is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
min L1 is V11() real ext-real Element of REAL
min_p L1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
L1 . (min_p L1) is V11() real ext-real Element of REAL
R1 is V11() real ext-real Element of REAL
len L1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
L1 /. (min_p L1) is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
|.r.| is V11() real ext-real Element of REAL
|.r.| " is V11() real ext-real Element of REAL
N2 /. r is Element of the carrier of (REAL-NS n)
N2 . r is Element of the carrier of (REAL-NS n)
||.(N2 /. r).|| is V11() real ext-real non negative Element of REAL
(|.r.| ") * ||.(N2 /. r).|| is V11() real ext-real Element of REAL
|.r.| * (((sqrt n) ") * R2) is V11() real ext-real Element of REAL
(|.r.| * (((sqrt n) ") * R2)) ^2 is V11() real ext-real Element of REAL
K37((|.r.| * (((sqrt n) ") * R2)),(|.r.| * (((sqrt n) ") * R2))) is V11() real ext-real set
n |-> ((|.r.| * (((sqrt n) ") * R2)) ^2) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of n -tuples_on REAL
p0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
sqr p0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of n -tuples_on REAL
p0 (#) sqrreal is Relation-like Function-like complex-valued ext-real-valued real-valued set
p0 (#) p0 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K450() .: (p0,p0) is set
q is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of n -tuples_on REAL
L0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of n -tuples_on REAL
N3 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
q . N3 is V11() real ext-real Element of REAL
L0 . N3 is V11() real ext-real Element of REAL
R0 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
dom p0 is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
(sqr p0) . R0 is V11() real ext-real Element of REAL
p0 . R0 is V11() real ext-real Element of REAL
sqrreal . (p0 . R0) is V11() real ext-real Element of REAL
(p0 . R0) ^2 is V11() real ext-real Element of REAL
K37((p0 . R0),(p0 . R0)) is V11() real ext-real set
L1 . R0 is V11() real ext-real Element of REAL
L1 /. R0 is V11() real ext-real Element of REAL
Proj (R0,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (R0,n)) * N2 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
((Proj (R0,n)) * N2) /. r is Element of the carrier of (REAL-NS 1)
((Proj (R0,n)) * N2) . r is Element of the carrier of (REAL-NS 1)
||.(((Proj (R0,n)) * N2) /. r).|| is V11() real ext-real non negative Element of REAL
(|.r.| ") * ||.(((Proj (R0,n)) * N2) /. r).|| is V11() real ext-real Element of REAL
(((sqrt n) ") * R2) / (|.r.| ") is V11() real ext-real Element of REAL
K39((|.r.| ")) is V11() real ext-real set
K37((((sqrt n) ") * R2),K39((|.r.| "))) is V11() real ext-real set
<*(p0 . R0)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,(p0 . R0)] is set
{1,(p0 . R0)} is non empty V160() V161() V162() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,(p0 . R0)},{1}} is non empty set
{[1,(p0 . R0)]} is non empty Function-like set
(Proj (R0,n)) . (N2 . r) is Element of the carrier of (REAL-NS 1)
proj (R0,n) is non empty Relation-like REAL n -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
K7((REAL n),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL n),REAL)) is set
(proj (R0,n)) . p0 is V11() real ext-real Element of REAL
<*((proj (R0,n)) . p0)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,((proj (R0,n)) . p0)] is set
{1,((proj (R0,n)) . p0)} is non empty V160() V161() V162() set
{{1,((proj (R0,n)) . p0)},{1}} is non empty set
{[1,((proj (R0,n)) . p0)]} is non empty Function-like set
||.(((Proj (R0,n)) * N2) . r).|| is V11() real ext-real non negative Element of REAL
N is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
|.N.| is V11() real ext-real non negative Element of REAL
sqr N is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
N (#) sqrreal is Relation-like Function-like complex-valued ext-real-valued real-valued set
N (#) N is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K450() .: (N,N) is set
Sum (sqr N) is V11() real ext-real Element of REAL
sqrt (Sum (sqr N)) is V11() real ext-real Element of REAL
|.(p0 . R0).| is V11() real ext-real Element of REAL
|.(p0 . R0).| ^2 is V11() real ext-real Element of REAL
K37(|.(p0 . R0).|,|.(p0 . R0).|) is V11() real ext-real set
p0 . N3 is V11() real ext-real Element of REAL
(p0 . N3) ^2 is V11() real ext-real Element of REAL
K37((p0 . N3),(p0 . N3)) is V11() real ext-real set
dom (sqr p0) is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
N3 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
(sqr p0) . N3 is V11() real ext-real Element of REAL
sqrreal * p0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
dom (sqrreal * p0) is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
dom p0 is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
p0 . N3 is V11() real ext-real Element of REAL
sqrreal . (p0 . N3) is V11() real ext-real Element of REAL
(p0 . N3) ^2 is V11() real ext-real Element of REAL
K37((p0 . N3),(p0 . N3)) is V11() real ext-real set
q . 1 is V11() real ext-real Element of REAL
L0 . 1 is V11() real ext-real Element of REAL
N3 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
L0 . N3 is V11() real ext-real Element of REAL
q . N3 is V11() real ext-real Element of REAL
Sum q is V11() real ext-real Element of REAL
Sum L0 is V11() real ext-real Element of REAL
N3 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative set
q . N3 is V11() real ext-real Element of REAL
L0 . N3 is V11() real ext-real Element of REAL
n * ((|.r.| * (((sqrt n) ") * R2)) ^2) is V11() real ext-real Element of REAL
Sum (sqr p0) is V11() real ext-real Element of REAL
sqrt (n * ((|.r.| * (((sqrt n) ") * R2)) ^2)) is V11() real ext-real Element of REAL
|.p0.| is V11() real ext-real non negative Element of REAL
sqr p0 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
Sum (sqr p0) is V11() real ext-real Element of REAL
sqrt (Sum (sqr p0)) is V11() real ext-real Element of REAL
sqrt ((|.r.| * (((sqrt n) ") * R2)) ^2) is V11() real ext-real Element of REAL
(sqrt n) * (sqrt ((|.r.| * (((sqrt n) ") * R2)) ^2)) is V11() real ext-real Element of REAL
|.(|.r.| * (((sqrt n) ") * R2)).| is V11() real ext-real Element of REAL
(sqrt n) * |.(|.r.| * (((sqrt n) ") * R2)).| is V11() real ext-real Element of REAL
(sqrt n) * (|.r.| * (((sqrt n) ") * R2)) is V11() real ext-real Element of REAL
(sqrt n) * (((sqrt n) ") * R2) is V11() real ext-real Element of REAL
((sqrt n) * (((sqrt n) ") * R2)) * |.r.| is V11() real ext-real Element of REAL
|.p0.| / |.r.| is V11() real ext-real Element of REAL
K39(|.r.|) is V11() real ext-real set
K37(|.p0.|,K39(|.r.|)) is V11() real ext-real set
(sqrt n) * ((sqrt n) ") is V11() real ext-real Element of REAL
((sqrt n) * ((sqrt n) ")) * R2 is V11() real ext-real Element of REAL
1 * R2 is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
|.r.| is V11() real ext-real Element of REAL
|.r.| " is V11() real ext-real Element of REAL
N2 /. r is Element of the carrier of (REAL-NS n)
N2 . r is Element of the carrier of (REAL-NS n)
||.(N2 /. r).|| is V11() real ext-real non negative Element of REAL
(|.r.| ") * ||.(N2 /. r).|| is V11() real ext-real Element of REAL
x0 - (min S) is V11() real ext-real Element of REAL
K38((min S)) is V11() real ext-real set
K36(x0,K38((min S))) is V11() real ext-real set
x0 + (min S) is V11() real ext-real Element of REAL
].(x0 - (min S)),(x0 + (min S)).[ is V160() V161() V162() open Element of K6(REAL)
L1 is set
L2 is V160() V161() V162() open Neighbourhood of x0
R1 is V11() real ext-real Element of REAL
R1 - x0 is V11() real ext-real Element of REAL
K38(x0) is V11() real ext-real set
K36(R1,K38(x0)) is V11() real ext-real set
|.(R1 - x0).| is V11() real ext-real Element of REAL
S /. 1 is V11() real ext-real Element of REAL
x0 - (S /. 1) is V11() real ext-real Element of REAL
K38((S /. 1)) is V11() real ext-real set
K36(x0,K38((S /. 1))) is V11() real ext-real set
x0 + (S /. 1) is V11() real ext-real Element of REAL
].(x0 - (S /. 1)),(x0 + (S /. 1)).[ is V160() V161() V162() open Element of K6(REAL)
Proj (1,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (1,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
dom ((Proj (1,n)) * m) is V160() V161() V162() Element of K6(REAL)
S . 1 is V11() real ext-real Element of REAL
f is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
R2 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
L1 is V11() real ext-real Element of REAL
m /. L1 is Element of the carrier of (REAL-NS n)
(m /. L1) - (m /. x0) is Element of the carrier of (REAL-NS n)
- (m /. x0) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(m /. L1),(- (m /. x0))) is Element of the carrier of (REAL-NS n)
L1 - x0 is V11() real ext-real Element of REAL
K36(L1,K38(x0)) is V11() real ext-real set
f . (L1 - x0) is Element of the carrier of (REAL-NS n)
R2 /. (L1 - x0) is Element of the carrier of (REAL-NS n)
(f . (L1 - x0)) + (R2 /. (L1 - x0)) is Element of the carrier of (REAL-NS n)
dom R2 is V160() V161() V162() Element of K6(REAL)
R2 . (L1 - x0) is set
(L1 - x0) + x0 is V11() real ext-real Element of REAL
m /. ((L1 - x0) + x0) is Element of the carrier of (REAL-NS n)
(m /. ((L1 - x0) + x0)) - (m /. x0) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(m /. ((L1 - x0) + x0)),(- (m /. x0))) is Element of the carrier of (REAL-NS n)
N1 . (L1 - x0) is Element of the carrier of (REAL-NS n)
((m /. ((L1 - x0) + x0)) - (m /. x0)) - (N1 . (L1 - x0)) is Element of the carrier of (REAL-NS n)
- (N1 . (L1 - x0)) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),((m /. ((L1 - x0) + x0)) - (m /. x0)),(- (N1 . (L1 - x0)))) is Element of the carrier of (REAL-NS n)
- (f . (L1 - x0)) is Element of the carrier of (REAL-NS n)
((m /. L1) - (m /. x0)) + (- (f . (L1 - x0))) is Element of the carrier of (REAL-NS n)
(- (f . (L1 - x0))) + (f . (L1 - x0)) is Element of the carrier of (REAL-NS n)
((m /. L1) - (m /. x0)) + ((- (f . (L1 - x0))) + (f . (L1 - x0))) is Element of the carrier of (REAL-NS n)
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
((m /. L1) - (m /. x0)) + (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
L1 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
R1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
Proj (n,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
(Proj (n,m)) * x0 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
S is V11() real ext-real set
(m,x0,S) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(m) FinSequence-like Element of REAL m
(Proj (n,m)) . (m,x0,S) is set
diff (((Proj (n,m)) * x0),S) is Element of the carrier of (REAL-NS 1)
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
diff (T,S) is Element of the carrier of (REAL-NS m)
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
x0 is V11() real ext-real set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (S,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (S,n)) * m is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
S is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (T,n) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS 1))) is set
(Proj (T,n)) * S is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
n is set
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
x0 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS x0 is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS x0) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS x0)) is set
K6(K7(REAL, the carrier of (REAL-NS x0))) is set
Proj (m,x0) is non empty Relation-like the carrier of (REAL-NS x0) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS x0), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS x0), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS x0), the carrier of (REAL-NS 1))) is set
S is Relation-like REAL -defined the carrier of (REAL-NS x0) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS x0)))
(Proj (m,x0)) * S is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
S `| n is Relation-like REAL -defined the carrier of (REAL-NS x0) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS x0)))
(Proj (m,x0)) * (S `| n) is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
((Proj (m,x0)) * S) `| n is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
dom (Proj (m,x0)) is Element of K6( the carrier of (REAL-NS x0))
K6( the carrier of (REAL-NS x0)) is set
rng S is Element of K6( the carrier of (REAL-NS x0))
dom ((Proj (m,x0)) * S) is V160() V161() V162() Element of K6(REAL)
dom S is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
dom (((Proj (m,x0)) * S) `| n) is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
(((Proj (m,x0)) * S) `| n) . T is set
diff (((Proj (m,x0)) * S),T) is Element of the carrier of (REAL-NS 1)
dom (S `| n) is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
(S `| n) . T is set
diff (S,T) is Element of the carrier of (REAL-NS x0)
rng (S `| n) is Element of K6( the carrier of (REAL-NS x0))
dom ((Proj (m,x0)) * (S `| n)) is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
diff (S,T) is Element of the carrier of (REAL-NS x0)
(Proj (m,x0)) . (diff (S,T)) is Element of the carrier of (REAL-NS 1)
diff (((Proj (m,x0)) * S),T) is Element of the carrier of (REAL-NS 1)
(((Proj (m,x0)) * S) `| n) . T is set
(S `| n) . T is set
((Proj (m,x0)) * (S `| n)) . T is set
n is set
m is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
x0 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL x0 is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
x0 -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL x0)) is set
K6(K7(REAL,(REAL x0))) is set
REAL-NS x0 is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS x0) is non empty non trivial set
Proj (m,x0) is non empty Relation-like the carrier of (REAL-NS x0) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS x0), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS x0), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS x0), the carrier of (REAL-NS 1))) is set
S is Relation-like REAL -defined REAL x0 -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL x0)))
(Proj (m,x0)) * S is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
(x0,S,n) is Relation-like REAL -defined REAL x0 -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL x0)))
(Proj (m,x0)) * (x0,S,n) is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
((Proj (m,x0)) * S) `| n is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS x0)) is set
K6(K7(REAL, the carrier of (REAL-NS x0))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS x0) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS x0)))
dom T is V160() V161() V162() Element of K6(REAL)
g is V11() real ext-real Element of REAL
T `| n is Relation-like REAL -defined the carrier of (REAL-NS x0) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS x0)))
dom (T `| n) is V160() V161() V162() Element of K6(REAL)
g is V11() real ext-real Element of REAL
(T `| n) . g is set
diff (T,g) is Element of the carrier of (REAL-NS x0)
dom (x0,S,n) is V160() V161() V162() Element of K6(REAL)
g is V11() real ext-real Element of REAL
(x0,S,n) . g is Relation-like Function-like complex-valued ext-real-valued real-valued set
(x0,S,g) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(x0) FinSequence-like Element of REAL x0
diff (T,g) is Element of the carrier of (REAL-NS x0)
(T `| n) . g is set
(Proj (m,x0)) * (T `| n) is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
n is set
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
x0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (S,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
(Proj (S,m)) * x0 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
Proj (1,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
(Proj (1,m)) * x0 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
dom (Proj (1,m)) is Element of K6( the carrier of (REAL-NS m))
K6( the carrier of (REAL-NS m)) is set
rng x0 is Element of K6( the carrier of (REAL-NS m))
dom ((Proj (1,m)) * x0) is V160() V161() V162() Element of K6(REAL)
dom x0 is V160() V161() V162() Element of K6(REAL)
S is V11() real ext-real Element of REAL
T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (T,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
(Proj (T,m)) * x0 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
n is set
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL m is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
m -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL m)) is set
K6(K7(REAL,(REAL m))) is set
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
x0 is Relation-like REAL -defined REAL m -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL m)))
S is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (S,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
(Proj (S,m)) * x0 is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
S is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
T is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
Proj (T,m) is non empty Relation-like the carrier of (REAL-NS m) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)))
K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1)) is set
K6(K7( the carrier of (REAL-NS m), the carrier of (REAL-NS 1))) is set
(Proj (T,m)) * S is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
dom x0 is V160() V161() V162() Element of K6(REAL)
T is V11() real ext-real Element of REAL
K7( the carrier of (REAL-NS 1),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS 1),REAL)) is set
n is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
m is Element of the carrier of (REAL-NS 1)
dom n is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
x0 is V11() real ext-real Element of REAL
S is V11() real ext-real Element of REAL
n /. m is V11() real ext-real Element of REAL
n . m is V11() real ext-real Element of REAL
T is Element of the carrier of (REAL-NS 1)
T - m is Element of the carrier of (REAL-NS 1)
- m is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),T,(- m)) is Element of the carrier of (REAL-NS 1)
||.(T - m).|| is V11() real ext-real non negative Element of REAL
n /. T is V11() real ext-real Element of REAL
n . T is V11() real ext-real Element of REAL
(n /. T) - (n /. m) is V11() real ext-real Element of REAL
K38((n /. m)) is V11() real ext-real set
K36((n /. T),K38((n /. m))) is V11() real ext-real set
abs ((n /. T) - (n /. m)) is V11() real ext-real Element of REAL
n . (T - m) is V11() real ext-real Element of REAL
K7(REAL, the carrier of (REAL-NS 1)) is set
K6(K7(REAL, the carrier of (REAL-NS 1))) is set
n is V11() real ext-real Element of REAL
m is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
dom m is V160() V161() V162() Element of K6(REAL)
S is V11() real ext-real Element of REAL
T is V11() real ext-real set
g is V11() real ext-real set
x0 is V11() real ext-real set
m /. x0 is Element of the carrier of (REAL-NS 1)
N1 is V11() real ext-real set
N1 - x0 is V11() real ext-real Element of REAL
K38(x0) is V11() real ext-real set
K36(N1,K38(x0)) is V11() real ext-real set
abs (N1 - x0) is V11() real ext-real Element of REAL
m /. N1 is Element of the carrier of (REAL-NS 1)
(m /. N1) - (m /. x0) is Element of the carrier of (REAL-NS 1)
- (m /. x0) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(m /. N1),(- (m /. x0))) is Element of the carrier of (REAL-NS 1)
||.((m /. N1) - (m /. x0)).|| is V11() real ext-real non negative Element of REAL
f is V11() real ext-real Element of REAL
m /. f is Element of the carrier of (REAL-NS 1)
m . f is Element of the carrier of (REAL-NS 1)
m /. n is Element of the carrier of (REAL-NS 1)
m . n is Element of the carrier of (REAL-NS 1)
(m /. f) - (m /. n) is Element of the carrier of (REAL-NS 1)
- (m /. n) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(m /. f),(- (m /. n))) is Element of the carrier of (REAL-NS 1)
f - n is V11() real ext-real Element of REAL
K38(n) is V11() real ext-real set
K36(f,K38(n)) is V11() real ext-real set
m . (f - n) is Element of the carrier of (REAL-NS 1)
n is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of n is non empty set
K7( the carrier of n,REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of n,REAL)) is set
m is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like NORMSTR
the carrier of m is non empty set
K7(REAL, the carrier of m) is set
K6(K7(REAL, the carrier of m)) is set
x0 is Relation-like the carrier of n -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of n,REAL))
S is Relation-like REAL -defined the carrier of m -valued Function-like Element of K6(K7(REAL, the carrier of m))
S * x0 is Relation-like the carrier of n -defined the carrier of m -valued Function-like Element of K6(K7( the carrier of n, the carrier of m))
K7( the carrier of n, the carrier of m) is set
K6(K7( the carrier of n, the carrier of m)) is set
dom (S * x0) is Element of K6( the carrier of n)
K6( the carrier of n) is set
T is Element of the carrier of n
x0 /. T is V11() real ext-real Element of REAL
K7(NAT, the carrier of n) is set
K6(K7(NAT, the carrier of n)) is set
(S * x0) /. T is Element of the carrier of m
g is non empty Relation-like NAT -defined the carrier of n -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of n))
rng g is Element of K6( the carrier of n)
lim g is Element of the carrier of n
(S * x0) /* g is non empty Relation-like NAT -defined the carrier of m -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of m))
K7(NAT, the carrier of m) is set
K6(K7(NAT, the carrier of m)) is set
lim ((S * x0) /* g) is Element of the carrier of m
dom x0 is Element of K6( the carrier of n)
N1 is set
x0 /* g is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
rng (x0 /* g) is V160() V161() V162() Element of K6(REAL)
f is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
(x0 /* g) . f is V11() real ext-real Element of REAL
g . f is Element of the carrier of n
x0 . (g . f) is V11() real ext-real Element of REAL
dom S is V160() V161() V162() Element of K6(REAL)
N1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
g . N1 is Element of the carrier of n
((S * x0) /* g) . N1 is Element of the carrier of m
(S * x0) . (g . N1) is set
x0 . (g . N1) is V11() real ext-real Element of REAL
S . (x0 . (g . N1)) is set
(x0 /* g) . N1 is V11() real ext-real Element of REAL
S . ((x0 /* g) . N1) is set
S /* (x0 /* g) is non empty Relation-like NAT -defined the carrier of m -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of m))
(S /* (x0 /* g)) . N1 is Element of the carrier of m
lim (x0 /* g) is V11() real ext-real Element of REAL
S /. (x0 /. T) is Element of the carrier of m
lim (S /* (x0 /* g)) is Element of the carrier of m
dom (proj (1,1)) is functional V233() V234() V235() Element of K6((REAL 1))
K6((REAL 1)) is set
rng (proj (1,1)) is V160() V161() V162() Element of K6(REAL)
m is set
x0 is V11() real ext-real Element of REAL
<*x0*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,x0] is set
{1,x0} is non empty V160() V161() V162() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,x0},{1}} is non empty set
{[1,x0]} is non empty Function-like set
S is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
(proj (1,1)) . S is V11() real ext-real Element of REAL
S . 1 is V11() real ext-real Element of REAL
m is V11() real ext-real Element of REAL
<*m*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,m] is set
{1,m} is non empty V160() V161() V162() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,m},{1}} is non empty set
{[1,m]} is non empty Function-like set
(proj (1,1)) . <*m*> is V11() real ext-real Element of REAL
x0 is V11() real ext-real Element of REAL
((proj (1,1)) ") . x0 is set
<*x0*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,x0] is set
{1,x0} is non empty V160() V161() V162() set
{{1,x0},{1}} is non empty set
{[1,x0]} is non empty Function-like set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))) is set
m is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
x0 is Element of the carrier of (REAL-NS 1)
S is V11() real ext-real Element of REAL
<*S*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,S] is set
{1,S} is non empty V160() V161() V162() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,S},{1}} is non empty set
{[1,S]} is non empty Function-like set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom T is V160() V161() V162() Element of K6(REAL)
T * m is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
g is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
dom g is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
f is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL 1)))
N1 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
m * N1 is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
g * N1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T * (id REAL) is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
N1 /. S is Element of the carrier of (REAL-NS 1)
N1 . S is Element of the carrier of (REAL-NS 1)
m /. x0 is V11() real ext-real Element of REAL
m . x0 is V11() real ext-real Element of REAL
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))) is set
m is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
x0 is Element of the carrier of (REAL-NS 1)
S is V11() real ext-real Element of REAL
<*S*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,S] is set
{1,S} is non empty V160() V161() V162() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,S},{1}} is non empty set
{[1,S]} is non empty Function-like set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom T is V160() V161() V162() Element of K6(REAL)
g is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
dom g is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
g * m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
m /. S is Element of the carrier of (REAL-NS 1)
m . S is Element of the carrier of (REAL-NS 1)
N1 is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
m * N1 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
id (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
id the carrier of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like one-to-one total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
T * N1 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
g * (id (REAL-NS 1)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
N1 /. x0 is V11() real ext-real Element of REAL
N1 . x0 is V11() real ext-real Element of REAL
<*1*> is Relation-like NAT -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued FinSequence-like FinSequence of NAT
[1,1] is set
{1,1} is non empty V160() V161() V162() V163() V164() V165() set
{1} is non empty V160() V161() V162() V163() V164() V165() set
{{1,1},{1}} is non empty set
{[1,1]} is non empty Function-like set
n is V11() real ext-real Element of REAL
m is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
diff (m,n) is Element of the carrier of (REAL-NS 1)
m . 1 is Element of the carrier of (REAL-NS 1)
x0 is Element of the carrier of (REAL-NS 1)
S is V11() real ext-real Element of REAL
m /. S is Element of the carrier of (REAL-NS 1)
m . S is Element of the carrier of (REAL-NS 1)
S * x0 is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),S,x0) is set
(S * x0) + (0. (REAL-NS 1)) is Element of the carrier of (REAL-NS 1)
T is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL 1)))
S * 1 is V11() real ext-real Element of REAL
<*(S * 1)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,(S * 1)] is set
{1,(S * 1)} is non empty V160() V161() V162() set
{{1,(S * 1)},{1}} is non empty set
{[1,(S * 1)]} is non empty Function-like set
S * <*1*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
K395(S) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (S,(id REAL)) is set
<*1*> (#) K395(S) is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom m is V160() V161() V162() Element of K6(REAL)
m `| ([#] REAL) is Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
S is V11() real ext-real Element of REAL
(m `| ([#] REAL)) . S is set
(m `| ([#] REAL)) . n is set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7( the carrier of (REAL-NS n),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS n),REAL)) is set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7( the carrier of (REAL-NS n),REAL) is complex-valued ext-real-valued real-valued set
K6(K7( the carrier of (REAL-NS n),REAL)) is set
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL n),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((REAL n),REAL)) is set
m is Relation-like the carrier of (REAL-NS n) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS n),REAL))
x0 is Element of the carrier of (REAL-NS n)
S is Relation-like REAL n -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
T is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
diff (S,T) is non empty Relation-like REAL n -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
<>* S is Relation-like REAL n -defined REAL 1 -valued Function-like V239() V240() V241() Element of K6(K7((REAL n),(REAL 1)))
K7((REAL n),(REAL 1)) is set
K6(K7((REAL n),(REAL 1))) is set
S (#) ((proj (1,1)) ") is Relation-like Function-like set
diff ((<>* S),T) is non empty Relation-like REAL n -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL n),(REAL 1)))
(proj (1,1)) * (diff ((<>* S),T)) is non empty Relation-like REAL n -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
g is Relation-like REAL n -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
N1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
S is non empty Relation-like the carrier of (REAL-NS n) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS n),REAL))
diff (g,N1) is non empty Relation-like REAL n -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
<>* g is Relation-like REAL n -defined REAL 1 -valued Function-like V239() V240() V241() Element of K6(K7((REAL n),(REAL 1)))
K7((REAL n),(REAL 1)) is set
K6(K7((REAL n),(REAL 1))) is set
g (#) ((proj (1,1)) ") is Relation-like Function-like set
diff ((<>* g),N1) is non empty Relation-like REAL n -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL n),(REAL 1)))
(proj (1,1)) * (diff ((<>* g),N1)) is non empty Relation-like REAL n -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
f is Relation-like REAL n -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
N2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(n) FinSequence-like Element of REAL n
T is non empty Relation-like the carrier of (REAL-NS n) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS n),REAL))
diff (f,N2) is non empty Relation-like REAL n -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
<>* f is Relation-like REAL n -defined REAL 1 -valued Function-like V239() V240() V241() Element of K6(K7((REAL n),(REAL 1)))
f (#) ((proj (1,1)) ") is Relation-like Function-like set
diff ((<>* f),N2) is non empty Relation-like REAL n -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL n),(REAL 1)))
(proj (1,1)) * (diff ((<>* f),N2)) is non empty Relation-like REAL n -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL n),REAL))
n is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
diff (n,m) is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
<>* n is Relation-like REAL 1 -defined REAL 1 -valued Function-like V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
n (#) ((proj (1,1)) ") is Relation-like Function-like set
diff ((<>* n),m) is non empty Relation-like REAL 1 -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
(proj (1,1)) * (diff ((<>* n),m)) is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
Seg 1 is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT : ( 1 <= b₁ & b₁ <= 1 ) } is set
0* 1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
1 |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of 1 -tuples_on REAL
(REAL 1) --> (0* 1) is non empty Relation-like REAL 1 -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
rng n is V160() V161() V162() Element of K6(REAL)
T is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL 1)))
dom (<>* n) is functional V233() V234() V235() Element of K6((REAL 1))
dom n is functional V233() V234() V235() Element of K6((REAL 1))
g is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL 1)))
T is non empty Relation-like REAL 1 -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
id (dom n) is Relation-like dom n -defined dom n -valued Function-like one-to-one total quasi_total V239() V240() V241() Element of K6(K7((dom n),(dom n)))
K7((dom n),(dom n)) is set
K6(K7((dom n),(dom n))) is set
id (REAL 1) is non empty Relation-like REAL 1 -defined REAL 1 -valued Function-like one-to-one total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
N1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
N1 + f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() .: (N1,f) is set
T . (N1 + f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
T . N1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
T . f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
(T . N1) + (T . f) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K448() .: ((T . N1),(T . f)) is set
(T . N1) + f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K448() .: ((T . N1),f) is set
N1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
T . N1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
f is V11() real ext-real Element of REAL
f * N1 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K395(f) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (f,(id REAL)) is set
N1 (#) K395(f) is Relation-like Function-like complex-valued ext-real-valued real-valued set
T . (f * N1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
f * (T . N1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
(T . N1) (#) K395(f) is Relation-like Function-like complex-valued ext-real-valued real-valued set
N1 is V11() real ext-real Element of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1 : not N1 <= |.(b₁ - m).| } is set
f is set
N2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
N2 - m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K449() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K446() is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K448() * ((id REAL),K446()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K449() .: (N2,m) is set
- m is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
m (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) m is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K395(K38(1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K450() [;] (K38(1),(id REAL)) is set
m (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
N2 + (- m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (N2,(- m)) is set
|.(N2 - m).| is V11() real ext-real non negative Element of REAL
sqr (N2 - m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
(N2 - m) (#) sqrreal is Relation-like Function-like complex-valued ext-real-valued real-valued set
(N2 - m) (#) (N2 - m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K450() .: ((N2 - m),(N2 - m)) is set
Sum (sqr (N2 - m)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (N2 - m))) is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
N2 is V11() real ext-real Element of REAL
R2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
|.R2.| is V11() real ext-real non negative Element of REAL
sqr R2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
R2 (#) sqrreal is Relation-like Function-like complex-valued ext-real-valued real-valued set
R2 (#) R2 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K450() .: (R2,R2) is set
Sum (sqr R2) is V11() real ext-real Element of REAL
sqrt (Sum (sqr R2)) is V11() real ext-real Element of REAL
((REAL 1) --> (0* 1)) . R2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
|.R2.| " is V11() real ext-real non negative Element of REAL
L2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
|.L2.| is V11() real ext-real non negative Element of REAL
sqr L2 is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
L2 (#) sqrreal is Relation-like Function-like complex-valued ext-real-valued real-valued set
L2 (#) L2 is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K450() .: (L2,L2) is set
Sum (sqr L2) is V11() real ext-real Element of REAL
sqrt (Sum (sqr L2)) is V11() real ext-real Element of REAL
(|.R2.| ") * |.L2.| is V11() real ext-real non negative Element of REAL
(<>* n) /. m is Relation-like Function-like complex-valued ext-real-valued real-valued V56(1) Element of REAL 1
f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
f - m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K449() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K448() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K446() is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K448() * ((id REAL),K446()) is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K449() .: (f,m) is set
- m is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
m (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) is V11() real ext-real non positive set
K38(1) (#) m is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K395(K38(1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
K450() [;] (K38(1),(id REAL)) is set
m (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
f + (- m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: (f,(- m)) is set
|.(f - m).| is V11() real ext-real non negative Element of REAL
sqr (f - m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
(f - m) (#) sqrreal is Relation-like Function-like complex-valued ext-real-valued real-valued set
(f - m) (#) (f - m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K450() .: ((f - m),(f - m)) is set
Sum (sqr (f - m)) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (f - m))) is V11() real ext-real Element of REAL
(<>* n) /. f is Relation-like Function-like complex-valued ext-real-valued real-valued V56(1) Element of REAL 1
((<>* n) /. f) - ((<>* n) /. m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
- ((<>* n) /. m) is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) (#) ((<>* n) /. m) is Relation-like Function-like complex-valued ext-real-valued real-valued set
((<>* n) /. f) + (- ((<>* n) /. m)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
T . (f - m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
((REAL 1) --> (0* 1)) . (f - m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
(T . (f - m)) + (((REAL 1) --> (0* 1)) . (f - m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K448() .: ((T . (f - m)),(((REAL 1) --> (0* 1)) . (f - m))) is set
T . m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
- 1 is V11() real ext-real non positive V132() V159() Element of INT
T . f is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
(T . f) - (T . m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K449() .: ((T . f),(T . m)) is set
- (T . m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
(T . m) (#) K446() is Relation-like Function-like complex-valued ext-real-valued real-valued set
K38(1) (#) (T . m) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
(T . m) (#) K395(K38(1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(T . f) + (- (T . m)) is Relation-like NAT -defined Function-like complex-valued ext-real-valued real-valued FinSequence-like set
K448() .: ((T . f),(- (T . m))) is set
(- 1) * m is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K395((- 1)) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() [;] ((- 1),(id REAL)) is set
m (#) K395((- 1)) is Relation-like Function-like complex-valued ext-real-valued real-valued set
T . ((- 1) * m) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
(T . f) + (T . ((- 1) * m)) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K448() .: ((T . f),(T . ((- 1) * m))) is set
(T . (f - m)) + (0* 1) is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
K448() .: ((T . (f - m)),(0* 1)) is set
(proj (1,1)) * T is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
((proj (1,1)) ") (#) (proj (1,1)) is Relation-like Function-like one-to-one complex-valued ext-real-valued real-valued set
n (#) (((proj (1,1)) ") (#) (proj (1,1))) is Relation-like Function-like complex-valued ext-real-valued real-valued set
id (rng (proj (1,1))) is Relation-like rng (proj (1,1)) -defined rng (proj (1,1)) -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7((rng (proj (1,1))),(rng (proj (1,1)))))
K7((rng (proj (1,1))),(rng (proj (1,1)))) is complex-valued ext-real-valued real-valued set
K6(K7((rng (proj (1,1))),(rng (proj (1,1))))) is set
(id (rng (proj (1,1)))) * n is Relation-like REAL 1 -defined rng (proj (1,1)) -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),(rng (proj (1,1)))))
K7((REAL 1),(rng (proj (1,1)))) is complex-valued ext-real-valued real-valued set
K6(K7((REAL 1),(rng (proj (1,1))))) is set
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
(id REAL) * n is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
n is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
m is Element of the carrier of (REAL-NS 1)
(1,n,m) is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
x0 is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
S is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
diff (x0,S) is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
<>* x0 is Relation-like REAL 1 -defined REAL 1 -valued Function-like V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
x0 (#) ((proj (1,1)) ") is Relation-like Function-like set
diff ((<>* x0),S) is non empty Relation-like REAL 1 -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
(proj (1,1)) * (diff ((<>* x0),S)) is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
T is Relation-like REAL 1 -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
g is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued V56(1) FinSequence-like Element of REAL 1
diff (T,g) is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
<>* T is Relation-like REAL 1 -defined REAL 1 -valued Function-like V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
T (#) ((proj (1,1)) ") is Relation-like Function-like set
diff ((<>* T),g) is non empty Relation-like REAL 1 -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
(proj (1,1)) * (diff ((<>* T),g)) is non empty Relation-like REAL 1 -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((REAL 1),REAL))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))) is set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
x0 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
x0 * m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL 1),(REAL n)) is set
K6(K7((REAL 1),(REAL n))) is set
S is non empty Relation-like REAL 1 -defined REAL n -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL n)))
S * m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom T is V160() V161() V162() Element of K6(REAL)
g is V11() real ext-real Element of REAL
N1 is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
|.f.| is V11() real ext-real Element of REAL
|.f.| " is V11() real ext-real Element of REAL
T /. f is Element of the carrier of (REAL-NS n)
||.(T /. f).|| is V11() real ext-real non negative Element of REAL
(|.f.| ") * ||.(T /. f).|| is V11() real ext-real Element of REAL
abs f is V11() real ext-real Element of REAL
m . f is Element of the carrier of (REAL-NS 1)
R2 is ext-real Element of ExtREAL
|.R2.| is ext-real Element of ExtREAL
N2 is Element of the carrier of (REAL-NS 1)
||.N2.|| is V11() real ext-real non negative Element of REAL
dom m is V160() V161() V162() Element of K6(REAL)
dom x0 is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
x0 /. N2 is Element of the carrier of (REAL-NS n)
x0 . (m . f) is set
T . f is set
||.(x0 /. N2).|| is V11() real ext-real non negative Element of REAL
||.N2.|| " is V11() real ext-real non negative Element of REAL
(abs f) " is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
|.f.| is V11() real ext-real Element of REAL
|.f.| " is V11() real ext-real Element of REAL
T /. f is Element of the carrier of (REAL-NS n)
||.(T /. f).|| is V11() real ext-real non negative Element of REAL
(|.f.| ") * ||.(T /. f).|| is V11() real ext-real Element of REAL
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
g is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
T /* g is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
K7(NAT, the carrier of (REAL-NS n)) is set
K6(K7(NAT, the carrier of (REAL-NS n))) is set
g " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(g ") (#) (T /* g) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
lim ((g ") (#) (T /* g)) is Element of the carrier of (REAL-NS n)
lim g is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() Element of REAL
N1 is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
N2 is V11() real ext-real set
R2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
L1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
g . L1 is V11() real ext-real Element of REAL
rng g is V160() V161() V162() Element of K6(REAL)
abs (g . L1) is V11() real ext-real Element of REAL
(abs (g . L1)) " is V11() real ext-real Element of REAL
T /. (g . L1) is Element of the carrier of (REAL-NS n)
||.(T /. (g . L1)).|| is V11() real ext-real non negative Element of REAL
((abs (g . L1)) ") * ||.(T /. (g . L1)).|| is V11() real ext-real Element of REAL
(T /* g) . L1 is Element of the carrier of (REAL-NS n)
||.((T /* g) . L1).|| is V11() real ext-real non negative Element of REAL
((abs (g . L1)) ") * ||.((T /* g) . L1).|| is V11() real ext-real Element of REAL
abs g is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(abs g) . L1 is V11() real ext-real Element of REAL
((abs g) . L1) " is V11() real ext-real Element of REAL
(((abs g) . L1) ") * ||.((T /* g) . L1).|| is V11() real ext-real Element of REAL
(abs g) " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
((abs g) ") . L1 is V11() real ext-real Element of REAL
(((abs g) ") . L1) * ||.((T /* g) . L1).|| is V11() real ext-real Element of REAL
abs (g ") is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(abs (g ")) . L1 is V11() real ext-real Element of REAL
((abs (g ")) . L1) * ||.((T /* g) . L1).|| is V11() real ext-real Element of REAL
(g ") . L1 is V11() real ext-real Element of REAL
abs ((g ") . L1) is V11() real ext-real Element of REAL
(abs ((g ") . L1)) * ||.((T /* g) . L1).|| is V11() real ext-real Element of REAL
((g ") . L1) * ((T /* g) . L1) is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),((g ") . L1),((T /* g) . L1)) is set
||.(((g ") . L1) * ((T /* g) . L1)).|| is V11() real ext-real non negative Element of REAL
((g ") (#) (T /* g)) . L1 is Element of the carrier of (REAL-NS n)
||.(((g ") (#) (T /* g)) . L1).|| is V11() real ext-real non negative Element of REAL
(((g ") (#) (T /* g)) . L1) - (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
- (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(((g ") (#) (T /* g)) . L1),(- (0. (REAL-NS n)))) is Element of the carrier of (REAL-NS n)
||.((((g ") (#) (T /* g)) . L1) - (0. (REAL-NS n))).|| is V11() real ext-real non negative Element of REAL
L2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
(g . L1) - 0 is V11() real ext-real Element of REAL
K38(0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() set
K36((g . L1),K38(0)) is V11() real ext-real set
abs ((g . L1) - 0) is V11() real ext-real Element of REAL
x0 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total V153( REAL-NS 1, REAL-NS n) V154( REAL-NS 1, REAL-NS n) Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
x0 * m is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS n)))
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7((REAL 1),(REAL n)) is set
K6(K7((REAL 1),(REAL n))) is set
S is non empty Relation-like REAL 1 -defined REAL n -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL n)))
S * m is Relation-like REAL -defined REAL n -valued Function-like V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T . 1 is set
dom (S * m) is V160() V161() V162() Element of K6(REAL)
g is Element of the carrier of (REAL-NS n)
m . 1 is Element of the carrier of (REAL-NS 1)
f is V11() real ext-real Element of REAL
T . f is set
f * g is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),f,g) is set
dom m is V160() V161() V162() Element of K6(REAL)
f * 1 is V11() real ext-real Element of REAL
m . (f * 1) is Element of the carrier of (REAL-NS 1)
x0 . (m . (f * 1)) is Element of the carrier of (REAL-NS n)
N1 is Element of the carrier of (REAL-NS 1)
f * N1 is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),f,N1) is set
x0 . (f * N1) is Element of the carrier of (REAL-NS n)
x0 . N1 is Element of the carrier of (REAL-NS n)
f * (x0 . N1) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),f,(x0 . N1)) is set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))) is set
m is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
x0 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
x0 * m is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
S is non empty Relation-like REAL -defined REAL n -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL n)))
S * m is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL n -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7( the carrier of (REAL-NS 1),(REAL n)))
K7( the carrier of (REAL-NS 1),(REAL n)) is set
K6(K7( the carrier of (REAL-NS 1),(REAL n))) is set
K7(NAT, the carrier of (REAL-NS 1)) is set
K6(K7(NAT, the carrier of (REAL-NS 1))) is set
T is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
g is non empty Relation-like NAT -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total non-zero 0. (REAL-NS 1) -convergent Element of K6(K7(NAT, the carrier of (REAL-NS 1)))
T /* g is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
K7(NAT, the carrier of (REAL-NS n)) is set
K6(K7(NAT, the carrier of (REAL-NS n))) is set
||.g.|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.g.|| " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(||.g.|| ") (#) (T /* g) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
lim ((||.g.|| ") (#) (T /* g)) is Element of the carrier of (REAL-NS n)
lim g is Element of the carrier of (REAL-NS 1)
N1 is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
f is V11() real ext-real set
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
R2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
L2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
g . L2 is Element of the carrier of (REAL-NS 1)
(g . L2) - (0. (REAL-NS 1)) is Element of the carrier of (REAL-NS 1)
- (0. (REAL-NS 1)) is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(g . L2),(- (0. (REAL-NS 1)))) is Element of the carrier of (REAL-NS 1)
||.((g . L2) - (0. (REAL-NS 1))).|| is V11() real ext-real non negative Element of REAL
||.(g . L2).|| is V11() real ext-real non negative Element of REAL
N1 . L2 is V11() real ext-real Element of REAL
m . (g . L2) is V11() real ext-real Element of REAL
(N1 . L2) - 0 is V11() real ext-real Element of REAL
K38(0) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() set
K36((N1 . L2),K38(0)) is V11() real ext-real set
abs ((N1 . L2) - 0) is V11() real ext-real Element of REAL
L2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
N1 . L2 is V11() real ext-real Element of REAL
(N1 . L2) - 0 is V11() real ext-real Element of REAL
K36((N1 . L2),K38(0)) is V11() real ext-real set
abs ((N1 . L2) - 0) is V11() real ext-real Element of REAL
lim N1 is V11() real ext-real Element of REAL
f is set
N2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
N1 . N2 is V11() real ext-real Element of REAL
abs (N1 . N2) is V11() real ext-real Element of REAL
g . N2 is Element of the carrier of (REAL-NS 1)
||.(g . N2).|| is V11() real ext-real non negative Element of REAL
m . (g . N2) is V11() real ext-real Element of REAL
N1 . f is V11() real ext-real Element of REAL
rng g is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
f is non empty Relation-like non-empty NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued 0 -convergent convergent Element of K6(K7(NAT,REAL))
x0 /* f is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
R2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
(x0 /* f) . R2 is Element of the carrier of (REAL-NS n)
f . R2 is V11() real ext-real Element of REAL
x0 . (f . R2) is set
g . R2 is Element of the carrier of (REAL-NS 1)
m . (g . R2) is V11() real ext-real Element of REAL
x0 . (m . (g . R2)) is set
dom g is V160() V161() V162() V163() V164() V165() Element of K6(NAT)
T . (g . R2) is set
N2 is Relation-like Function-like set
g (#) N2 is Relation-like Function-like set
(g (#) N2) . R2 is set
dom T is Element of K6( the carrier of (REAL-NS 1))
(T /* g) . R2 is Element of the carrier of (REAL-NS n)
||.((||.g.|| ") (#) (T /* g)).|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.((||.g.|| ") (#) (T /* g)).|| . R2 is V11() real ext-real Element of REAL
((||.g.|| ") (#) (T /* g)) . R2 is Element of the carrier of (REAL-NS n)
||.(((||.g.|| ") (#) (T /* g)) . R2).|| is V11() real ext-real non negative Element of REAL
(||.g.|| ") . R2 is V11() real ext-real Element of REAL
((||.g.|| ") . R2) * ((T /* g) . R2) is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),((||.g.|| ") . R2),((T /* g) . R2)) is set
||.(((||.g.|| ") . R2) * ((T /* g) . R2)).|| is V11() real ext-real non negative Element of REAL
abs ((||.g.|| ") . R2) is V11() real ext-real Element of REAL
||.((T /* g) . R2).|| is V11() real ext-real non negative Element of REAL
(abs ((||.g.|| ") . R2)) * ||.((T /* g) . R2).|| is V11() real ext-real Element of REAL
||.g.|| . R2 is V11() real ext-real Element of REAL
(||.g.|| . R2) " is V11() real ext-real Element of REAL
abs ((||.g.|| . R2) ") is V11() real ext-real Element of REAL
(abs ((||.g.|| . R2) ")) * ||.((T /* g) . R2).|| is V11() real ext-real Element of REAL
||.(g . R2).|| is V11() real ext-real non negative Element of REAL
||.(g . R2).|| " is V11() real ext-real non negative Element of REAL
abs (||.(g . R2).|| ") is V11() real ext-real Element of REAL
(abs (||.(g . R2).|| ")) * ||.((T /* g) . R2).|| is V11() real ext-real Element of REAL
(||.(g . R2).|| ") * ||.((T /* g) . R2).|| is V11() real ext-real non negative Element of REAL
abs (f . R2) is V11() real ext-real Element of REAL
(abs (f . R2)) " is V11() real ext-real Element of REAL
((abs (f . R2)) ") * ||.((T /* g) . R2).|| is V11() real ext-real Element of REAL
||.((x0 /* f) . R2).|| is V11() real ext-real non negative Element of REAL
((abs (f . R2)) ") * ||.((x0 /* f) . R2).|| is V11() real ext-real Element of REAL
abs f is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(abs f) . R2 is V11() real ext-real Element of REAL
((abs f) . R2) " is V11() real ext-real Element of REAL
(((abs f) . R2) ") * ||.((x0 /* f) . R2).|| is V11() real ext-real Element of REAL
(abs f) " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
((abs f) ") . R2 is V11() real ext-real Element of REAL
(((abs f) ") . R2) * ||.((x0 /* f) . R2).|| is V11() real ext-real Element of REAL
f " is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
abs (f ") is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
(abs (f ")) . R2 is V11() real ext-real Element of REAL
((abs (f ")) . R2) * ||.((x0 /* f) . R2).|| is V11() real ext-real Element of REAL
(f ") . R2 is V11() real ext-real Element of REAL
abs ((f ") . R2) is V11() real ext-real Element of REAL
(abs ((f ") . R2)) * ||.((x0 /* f) . R2).|| is V11() real ext-real Element of REAL
((f ") . R2) * ((x0 /* f) . R2) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),((f ") . R2),((x0 /* f) . R2)) is set
||.(((f ") . R2) * ((x0 /* f) . R2)).|| is V11() real ext-real non negative Element of REAL
(f ") (#) (x0 /* f) is non empty Relation-like NAT -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(NAT, the carrier of (REAL-NS n)))
((f ") (#) (x0 /* f)) . R2 is Element of the carrier of (REAL-NS n)
||.(((f ") (#) (x0 /* f)) . R2).|| is V11() real ext-real non negative Element of REAL
||.((f ") (#) (x0 /* f)).|| is non empty Relation-like NAT -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(NAT,REAL))
||.((f ") (#) (x0 /* f)).|| . R2 is V11() real ext-real Element of REAL
lim ((f ") (#) (x0 /* f)) is Element of the carrier of (REAL-NS n)
lim ||.((f ") (#) (x0 /* f)).|| is V11() real ext-real Element of REAL
||.(0. (REAL-NS n)).|| is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() Element of REAL
N2 is V11() real ext-real Element of REAL
R2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
L2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
L1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
||.((||.g.|| ") (#) (T /* g)).|| . L1 is V11() real ext-real Element of REAL
(||.((||.g.|| ") (#) (T /* g)).|| . L1) - 0 is V11() real ext-real Element of REAL
K36((||.((||.g.|| ") (#) (T /* g)).|| . L1),K38(0)) is V11() real ext-real set
abs ((||.((||.g.|| ") (#) (T /* g)).|| . L1) - 0) is V11() real ext-real Element of REAL
((||.g.|| ") (#) (T /* g)) . L1 is Element of the carrier of (REAL-NS n)
||.(((||.g.|| ") (#) (T /* g)) . L1).|| is V11() real ext-real non negative Element of REAL
abs ||.(((||.g.|| ") (#) (T /* g)) . L1).|| is V11() real ext-real Element of REAL
(((||.g.|| ") (#) (T /* g)) . L1) - (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
- (0. (REAL-NS n)) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(((||.g.|| ") (#) (T /* g)) . L1),(- (0. (REAL-NS n)))) is Element of the carrier of (REAL-NS n)
||.((((||.g.|| ") (#) (T /* g)) . L1) - (0. (REAL-NS n))).|| is V11() real ext-real non negative Element of REAL
x0 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
x0 * m is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
S is Element of the carrier of (REAL-NS n)
REAL n is non empty functional FinSequence-membered V233() V234() V235() FinSequenceSet of REAL
n -tuples_on REAL is non empty functional FinSequence-membered FinSequenceSet of REAL
K7(REAL,(REAL n)) is set
K6(K7(REAL,(REAL n))) is set
||.S.|| is V11() real ext-real non negative Element of REAL
N1 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
dom N1 is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
f is Element of the carrier of (REAL-NS 1)
N2 is Element of the carrier of (REAL-NS 1)
f + N2 is Element of the carrier of (REAL-NS 1)
N1 . (f + N2) is Element of the carrier of (REAL-NS n)
m . (f + N2) is V11() real ext-real Element of REAL
x0 . (m . (f + N2)) is Element of the carrier of (REAL-NS n)
m . f is V11() real ext-real Element of REAL
m . N2 is V11() real ext-real Element of REAL
(m . f) + (m . N2) is V11() real ext-real Element of REAL
x0 . ((m . f) + (m . N2)) is Element of the carrier of (REAL-NS n)
((m . f) + (m . N2)) * S is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),((m . f) + (m . N2)),S) is set
(m . f) * S is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),(m . f),S) is set
(m . N2) * S is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),(m . N2),S) is set
((m . f) * S) + ((m . N2) * S) is Element of the carrier of (REAL-NS n)
x0 . (m . f) is Element of the carrier of (REAL-NS n)
(x0 . (m . f)) + ((m . N2) * S) is Element of the carrier of (REAL-NS n)
x0 . (m . N2) is Element of the carrier of (REAL-NS n)
(x0 . (m . f)) + (x0 . (m . N2)) is Element of the carrier of (REAL-NS n)
N1 . f is Element of the carrier of (REAL-NS n)
N1 . N2 is Element of the carrier of (REAL-NS n)
(N1 . f) + (N1 . N2) is Element of the carrier of (REAL-NS n)
f is Element of the carrier of (REAL-NS 1)
N2 is V11() real ext-real Element of REAL
N2 * f is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),N2,f) is set
N1 . (N2 * f) is Element of the carrier of (REAL-NS n)
m . (N2 * f) is V11() real ext-real Element of REAL
x0 . (m . (N2 * f)) is Element of the carrier of (REAL-NS n)
m . f is V11() real ext-real Element of REAL
N2 * (m . f) is V11() real ext-real Element of REAL
x0 . (N2 * (m . f)) is Element of the carrier of (REAL-NS n)
(N2 * (m . f)) * S is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),(N2 * (m . f)),S) is set
(m . f) * S is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),(m . f),S) is set
N2 * ((m . f) * S) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),N2,((m . f) * S)) is set
x0 . (m . f) is Element of the carrier of (REAL-NS n)
N1 . f is Element of the carrier of (REAL-NS n)
N2 * (N1 . f) is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),N2,(N1 . f)) is set
f is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total V153( REAL-NS 1, REAL-NS n) V154( REAL-NS 1, REAL-NS n) Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
N2 is Element of the carrier of (REAL-NS 1)
f . N2 is Element of the carrier of (REAL-NS n)
||.(f . N2).|| is V11() real ext-real non negative Element of REAL
m . N2 is V11() real ext-real Element of REAL
x0 . (m . N2) is Element of the carrier of (REAL-NS n)
||.(x0 . (m . N2)).|| is V11() real ext-real non negative Element of REAL
(m . N2) * S is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),(m . N2),S) is set
||.((m . N2) * S).|| is V11() real ext-real non negative Element of REAL
|.(m . N2).| is V11() real ext-real Element of REAL
||.S.|| * |.(m . N2).| is V11() real ext-real Element of REAL
||.N2.|| is V11() real ext-real non negative Element of REAL
||.S.|| * ||.N2.|| is V11() real ext-real non negative Element of REAL
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))) is set
m is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
x0 is Element of the carrier of (REAL-NS 1)
S is V11() real ext-real Element of REAL
<*S*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,S] is set
{1,S} is non empty V160() V161() V162() set
{{1,S},{1}} is non empty set
{[1,S]} is non empty Function-like set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom T is V160() V161() V162() Element of K6(REAL)
diff (T,S) is Element of the carrier of (REAL-NS n)
g is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
dom g is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
g * m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
diff (g,x0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() NORMSTR
BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty functional Element of K6( the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n)))
K284( the carrier of (REAL-NS 1),(REAL-NS n)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS 1), the carrier of (REAL-NS n)
FuncZero ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS 1) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))))
K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))))) is set
FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(FuncZero ( the carrier of (REAL-NS 1),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS n))) #) is strict RLSStruct
the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n)) is non empty set
K6( the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n))) is set
K233(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS 1),(REAL-NS n))
K231(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
K232(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS n))),K233(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),K231(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),K232(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) #) is V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K233((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))
K231((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
K232((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),K233((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),K231((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),K232((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS n))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))) is non empty set
(diff (g,x0)) . <*1*> is set
g /. x0 is Element of the carrier of (REAL-NS n)
f is Neighbourhood of x0
N1 is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
m * N1 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
id (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
id the carrier of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like one-to-one total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
T * N1 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
g * (id (REAL-NS 1)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
N2 is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS 1) : not N2 <= ||.(b₁ - x0).|| } is set
R2 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))
L2 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
R2 is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))
L2 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
L2 * m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
m (#) R2 is Relation-like Function-like set
p0 is set
q is Element of the carrier of (REAL-NS 1)
q - x0 is Element of the carrier of (REAL-NS 1)
- x0 is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),q,(- x0)) is Element of the carrier of (REAL-NS 1)
||.(q - x0).|| is V11() real ext-real non negative Element of REAL
{ b₁ where b₁ is V11() real ext-real Element of REAL : not N2 <= abs (b₁ - S) } is set
p0 is Neighbourhood of x0
L0 is set
N3 is V11() real ext-real Element of REAL
N3 - S is V11() real ext-real Element of REAL
K38(S) is V11() real ext-real set
K36(N3,K38(S)) is V11() real ext-real set
abs (N3 - S) is V11() real ext-real Element of REAL
m . N3 is Element of the carrier of (REAL-NS 1)
m . S is Element of the carrier of (REAL-NS 1)
R0 is Element of the carrier of (REAL-NS 1)
R0 - x0 is Element of the carrier of (REAL-NS 1)
- x0 is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),R0,(- x0)) is Element of the carrier of (REAL-NS 1)
m . (N3 - S) is Element of the carrier of (REAL-NS 1)
||.(R0 - x0).|| is V11() real ext-real non negative Element of REAL
N1 . (m . N3) is V11() real ext-real Element of REAL
N1 .: p0 is V160() V161() V162() Element of K6(REAL)
N3 is set
N1 . N3 is V11() real ext-real Element of REAL
R0 is Element of the carrier of (REAL-NS 1)
R0 - x0 is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),R0,(- x0)) is Element of the carrier of (REAL-NS 1)
||.(R0 - x0).|| is V11() real ext-real non negative Element of REAL
N1 . R0 is V11() real ext-real Element of REAL
N1 . x0 is V11() real ext-real Element of REAL
N1 . (R0 - x0) is V11() real ext-real Element of REAL
N is V11() real ext-real Element of REAL
N - S is V11() real ext-real Element of REAL
K36(N,K38(S)) is V11() real ext-real set
abs (N - S) is V11() real ext-real Element of REAL
N1 .: (dom g) is V160() V161() V162() Element of K6(REAL)
N1 " (dom T) is Element of K6( the carrier of (REAL-NS 1))
N1 .: (N1 " (dom T)) is V160() V161() V162() Element of K6(REAL)
dom (g * m) is V160() V161() V162() Element of K6(REAL)
id (REAL 1) is non empty Relation-like REAL 1 -defined REAL 1 -valued Function-like one-to-one total quasi_total V239() V240() V241() Element of K6(K7((REAL 1),(REAL 1)))
S - N2 is V11() real ext-real Element of REAL
K38(N2) is V11() real ext-real set
K36(S,K38(N2)) is V11() real ext-real set
S + N2 is V11() real ext-real Element of REAL
].(S - N2),(S + N2).[ is V160() V161() V162() open Element of K6(REAL)
L0 is set
N3 is V11() real ext-real Element of REAL
N3 - S is V11() real ext-real Element of REAL
K36(N3,K38(S)) is V11() real ext-real set
abs (N3 - S) is V11() real ext-real Element of REAL
L0 is set
N3 is V11() real ext-real Element of REAL
N3 - S is V11() real ext-real Element of REAL
K36(N3,K38(S)) is V11() real ext-real set
abs (N3 - S) is V11() real ext-real Element of REAL
(m * N1) .: p0 is Element of K6( the carrier of (REAL-NS 1))
m .: { b₁ where b₁ is V11() real ext-real Element of REAL : not N2 <= abs (b₁ - S) } is Element of K6( the carrier of (REAL-NS 1))
(g * m) /. S is Element of the carrier of (REAL-NS n)
R1 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
L1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
L0 is V11() real ext-real Element of REAL
(g * m) /. L0 is Element of the carrier of (REAL-NS n)
((g * m) /. L0) - ((g * m) /. S) is Element of the carrier of (REAL-NS n)
- ((g * m) /. S) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),((g * m) /. L0),(- ((g * m) /. S))) is Element of the carrier of (REAL-NS n)
L0 - S is V11() real ext-real Element of REAL
K36(L0,K38(S)) is V11() real ext-real set
R1 . (L0 - S) is Element of the carrier of (REAL-NS n)
L1 /. (L0 - S) is Element of the carrier of (REAL-NS n)
(R1 . (L0 - S)) + (L1 /. (L0 - S)) is Element of the carrier of (REAL-NS n)
m . L0 is Element of the carrier of (REAL-NS 1)
dom L2 is Element of K6( the carrier of (REAL-NS 1))
dom (L2 * m) is V160() V161() V162() Element of K6(REAL)
m . (L0 - S) is Element of the carrier of (REAL-NS 1)
L2 /. (m . (L0 - S)) is Element of the carrier of (REAL-NS n)
L2 . (m . (L0 - S)) is set
L1 . (L0 - S) is set
dom (m (#) R2) is set
g /. (m . L0) is Element of the carrier of (REAL-NS n)
g . (m . L0) is set
(g * m) . L0 is set
g /. (m . S) is Element of the carrier of (REAL-NS n)
g . (m . S) is set
(g * m) . S is set
N3 is Element of the carrier of (REAL-NS 1)
g /. N3 is Element of the carrier of (REAL-NS n)
(g /. N3) - (g /. x0) is Element of the carrier of (REAL-NS n)
- (g /. x0) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(g /. N3),(- (g /. x0))) is Element of the carrier of (REAL-NS n)
N3 - x0 is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),N3,(- x0)) is Element of the carrier of (REAL-NS 1)
R2 . (N3 - x0) is Element of the carrier of (REAL-NS n)
L2 /. (N3 - x0) is Element of the carrier of (REAL-NS n)
(R2 . (N3 - x0)) + (L2 /. (N3 - x0)) is Element of the carrier of (REAL-NS n)
(g /. (m . L0)) - (g /. (m . S)) is Element of the carrier of (REAL-NS n)
- (g /. (m . S)) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(g /. (m . L0)),(- (g /. (m . S)))) is Element of the carrier of (REAL-NS n)
R2 . (m . (L0 - S)) is Element of the carrier of (REAL-NS n)
(R2 . (m . (L0 - S))) + (L2 /. (N3 - x0)) is Element of the carrier of (REAL-NS n)
diff ((g * m),S) is Element of the carrier of (REAL-NS n)
R1 . 1 is Element of the carrier of (REAL-NS n)
m (#) (diff (g,x0)) is Relation-like Function-like set
(m (#) (diff (g,x0))) . 1 is set
m . 1 is Element of the carrier of (REAL-NS 1)
(diff (g,x0)) . (m . 1) is Element of the carrier of (REAL-NS n)
L0 is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL 1)))
L0 is V11() real ext-real Element of REAL
<*L0*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,L0] is set
{1,L0} is non empty V160() V161() V162() set
{{1,L0},{1}} is non empty set
{[1,L0]} is non empty Function-like set
(diff (g,x0)) . <*L0*> is set
L0 * (diff (T,S)) is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),L0,(diff (T,S))) is set
L0 * 1 is V11() real ext-real Element of REAL
<*(L0 * 1)*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,(L0 * 1)] is set
{1,(L0 * 1)} is non empty V160() V161() V162() set
{{1,(L0 * 1)},{1}} is non empty set
{[1,(L0 * 1)]} is non empty Function-like set
(diff (g,x0)) . <*(L0 * 1)*> is set
L0 * <*1*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
K395(L0) is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
K450() is Relation-like K7(REAL,REAL) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(K7(REAL,REAL),REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
K450() [;] (L0,(id REAL)) is set
<*1*> (#) K395(L0) is Relation-like Function-like complex-valued ext-real-valued real-valued set
(diff (g,x0)) . (L0 * <*1*>) is set
N3 is Element of the carrier of (REAL-NS 1)
L0 * N3 is Element of the carrier of (REAL-NS 1)
the Mult of (REAL-NS 1) is Relation-like K7(REAL, the carrier of (REAL-NS 1)) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)))
K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS 1)), the carrier of (REAL-NS 1))) is set
K466( the Mult of (REAL-NS 1),L0,N3) is set
(diff (g,x0)) . (L0 * N3) is Element of the carrier of (REAL-NS n)
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))) is set
m is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
x0 is Element of the carrier of (REAL-NS 1)
S is V11() real ext-real Element of REAL
<*S*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,S] is set
{1,S} is non empty V160() V161() V162() set
{{1,S},{1}} is non empty set
{[1,S]} is non empty Function-like set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom T is V160() V161() V162() Element of K6(REAL)
g is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
dom g is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
g * m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom (g * m) is V160() V161() V162() Element of K6(REAL)
(g * m) /. S is Element of the carrier of (REAL-NS n)
f is V160() V161() V162() open Neighbourhood of S
N1 is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
m * N1 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
id (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
id the carrier of (REAL-NS 1) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS 1) -valued Function-like one-to-one total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS 1)))
T * N1 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
g * (id (REAL-NS 1)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
N2 is V11() real ext-real set
S - N2 is V11() real ext-real Element of REAL
K38(N2) is V11() real ext-real set
K36(S,K38(N2)) is V11() real ext-real set
S + N2 is V11() real ext-real Element of REAL
].(S - N2),(S + N2).[ is V160() V161() V162() open Element of K6(REAL)
R2 is V11() real ext-real Element of REAL
{ b₁ where b₁ is Element of the carrier of (REAL-NS 1) : not R2 <= ||.(b₁ - x0).|| } is set
L1 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
R1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
L1 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
R1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
R1 * N1 is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
L1 * N1 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
q is set
L0 is Element of the carrier of (REAL-NS 1)
L0 - x0 is Element of the carrier of (REAL-NS 1)
- x0 is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),L0,(- x0)) is Element of the carrier of (REAL-NS 1)
||.(L0 - x0).|| is V11() real ext-real non negative Element of REAL
N1 . L0 is V11() real ext-real Element of REAL
N1 . x0 is V11() real ext-real Element of REAL
N1 . (L0 - x0) is V11() real ext-real Element of REAL
N3 is V11() real ext-real Element of REAL
N3 - S is V11() real ext-real Element of REAL
K38(S) is V11() real ext-real set
K36(N3,K38(S)) is V11() real ext-real set
abs (N3 - S) is V11() real ext-real Element of REAL
dom N1 is Element of K6( the carrier of (REAL-NS 1))
m . N3 is Element of the carrier of (REAL-NS 1)
m .: f is Element of K6( the carrier of (REAL-NS 1))
L0 is set
m . L0 is set
N3 is V11() real ext-real Element of REAL
m . N3 is Element of the carrier of (REAL-NS 1)
m . S is Element of the carrier of (REAL-NS 1)
(m . N3) - x0 is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),(m . N3),(- x0)) is Element of the carrier of (REAL-NS 1)
N3 - S is V11() real ext-real Element of REAL
K36(N3,K38(S)) is V11() real ext-real set
m . (N3 - S) is Element of the carrier of (REAL-NS 1)
abs (N3 - S) is V11() real ext-real Element of REAL
||.((m . N3) - x0).|| is V11() real ext-real non negative Element of REAL
m .: (dom T) is Element of K6( the carrier of (REAL-NS 1))
m " (dom g) is V160() V161() V162() Element of K6(REAL)
m .: (m " (dom g)) is Element of K6( the carrier of (REAL-NS 1))
q is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL 1)))
q is set
L0 is Element of the carrier of (REAL-NS 1)
L0 - x0 is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),L0,(- x0)) is Element of the carrier of (REAL-NS 1)
||.(L0 - x0).|| is V11() real ext-real non negative Element of REAL
N1 * m is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
(N1 * m) .: f is V160() V161() V162() Element of K6(REAL)
N1 .: { b₁ where b₁ is Element of the carrier of (REAL-NS 1) : not R2 <= ||.(b₁ - x0).|| } is V160() V161() V162() Element of K6(REAL)
g /. x0 is Element of the carrier of (REAL-NS n)
p0 is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total V153( REAL-NS 1, REAL-NS n) V154( REAL-NS 1, REAL-NS n) Lipschitzian Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
r is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
T . (N1 . x0) is set
g . x0 is set
q is Element of the carrier of (REAL-NS 1)
g /. q is Element of the carrier of (REAL-NS n)
(g /. q) - (g /. x0) is Element of the carrier of (REAL-NS n)
- (g /. x0) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(g /. q),(- (g /. x0))) is Element of the carrier of (REAL-NS n)
q - x0 is Element of the carrier of (REAL-NS 1)
K192((REAL-NS 1),q,(- x0)) is Element of the carrier of (REAL-NS 1)
p0 . (q - x0) is Element of the carrier of (REAL-NS n)
r /. (q - x0) is Element of the carrier of (REAL-NS n)
(p0 . (q - x0)) + (r /. (q - x0)) is Element of the carrier of (REAL-NS n)
N1 . q is V11() real ext-real Element of REAL
T /. (N1 . q) is Element of the carrier of (REAL-NS n)
T /. S is Element of the carrier of (REAL-NS n)
(N1 . q) - S is V11() real ext-real Element of REAL
K36((N1 . q),K38(S)) is V11() real ext-real set
L1 . ((N1 . q) - S) is Element of the carrier of (REAL-NS n)
R1 /. ((N1 . q) - S) is Element of the carrier of (REAL-NS n)
(T /. (N1 . q)) - (T /. S) is Element of the carrier of (REAL-NS n)
- (T /. S) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(T /. (N1 . q)),(- (T /. S))) is Element of the carrier of (REAL-NS n)
N is Element of the carrier of (REAL-NS n)
x is Element of the carrier of (REAL-NS n)
N + x is Element of the carrier of (REAL-NS n)
T /. (N1 . x0) is Element of the carrier of (REAL-NS n)
(T /. (N1 . q)) - (T /. (N1 . x0)) is Element of the carrier of (REAL-NS n)
- (T /. (N1 . x0)) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(T /. (N1 . q)),(- (T /. (N1 . x0)))) is Element of the carrier of (REAL-NS n)
N1 . (q - x0) is V11() real ext-real Element of REAL
L1 . (N1 . (q - x0)) is Element of the carrier of (REAL-NS n)
(L1 . (N1 . (q - x0))) + (R1 /. ((N1 . q) - S)) is Element of the carrier of (REAL-NS n)
R1 /. (N1 . (q - x0)) is Element of the carrier of (REAL-NS n)
(L1 . (N1 . (q - x0))) + (R1 /. (N1 . (q - x0))) is Element of the carrier of (REAL-NS n)
dom p0 is Element of K6( the carrier of (REAL-NS 1))
R1 . (N1 . (q - x0)) is set
(R1 * N1) . (q - x0) is set
dom (R1 * N1) is Element of K6( the carrier of (REAL-NS 1))
dom R1 is V160() V161() V162() Element of K6(REAL)
T . (N1 . q) is set
g . q is set
R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() NORMSTR
BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty functional Element of K6( the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n)))
K284( the carrier of (REAL-NS 1),(REAL-NS n)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS 1), the carrier of (REAL-NS n)
FuncZero ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS 1) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))))
K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))))) is set
FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(FuncZero ( the carrier of (REAL-NS 1),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS n))) #) is strict RLSStruct
the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n)) is non empty set
K6( the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n))) is set
K233(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS 1),(REAL-NS n))
K231(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
K232(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS n))),K233(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),K231(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),K232(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) #) is V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K233((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))
K231((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
K232((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),K233((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),K231((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),K232((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS n))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))) is non empty set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))) is set
m is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
x0 is Element of the carrier of (REAL-NS 1)
S is V11() real ext-real Element of REAL
<*S*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,S] is set
{1,S} is non empty V160() V161() V162() set
{{1,S},{1}} is non empty set
{[1,S]} is non empty Function-like set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom T is V160() V161() V162() Element of K6(REAL)
T * m is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
g is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
dom g is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
f is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL 1)))
N1 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
m * N1 is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
g * N1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T * (id REAL) is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is set
K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))) is set
m is non empty Relation-like the carrier of (REAL-NS 1) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7( the carrier of (REAL-NS 1),REAL))
x0 is Element of the carrier of (REAL-NS 1)
S is V11() real ext-real Element of REAL
<*S*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,S] is set
{1,S} is non empty V160() V161() V162() set
{{1,S},{1}} is non empty set
{[1,S]} is non empty Function-like set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
dom T is V160() V161() V162() Element of K6(REAL)
T * m is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
diff (T,S) is Element of the carrier of (REAL-NS n)
g is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
dom g is Element of K6( the carrier of (REAL-NS 1))
K6( the carrier of (REAL-NS 1)) is set
diff (g,x0) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() NORMSTR
BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))))
R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
LinearOperators ((REAL-NS 1),(REAL-NS n)) is non empty functional Element of K6( the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n)))
K284( the carrier of (REAL-NS 1),(REAL-NS n)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS 1), the carrier of (REAL-NS n)
FuncZero ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
the carrier of (REAL-NS 1) --> (0. (REAL-NS n)) is non empty Relation-like the carrier of (REAL-NS 1) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))
FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))))
K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))))) is set
FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS n)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) -defined Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n)))),(Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS 1), the carrier of (REAL-NS n))),(FuncZero ( the carrier of (REAL-NS 1),(REAL-NS n))),(FuncAdd ( the carrier of (REAL-NS 1),(REAL-NS n))),(FuncExtMult ( the carrier of (REAL-NS 1),(REAL-NS n))) #) is strict RLSStruct
the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n)) is non empty set
K6( the carrier of K284( the carrier of (REAL-NS 1),(REAL-NS n))) is set
K233(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS 1),(REAL-NS n))
K231(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7((LinearOperators ((REAL-NS 1),(REAL-NS n))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
K232(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined LinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS 1),(REAL-NS n)))),(LinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
RLSStruct(# (LinearOperators ((REAL-NS 1),(REAL-NS n))),K233(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),K231(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))),K232(K284( the carrier of (REAL-NS 1),(REAL-NS n)),(LinearOperators ((REAL-NS 1),(REAL-NS n)))) #) is V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K233((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Element of BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))
K231((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
K232((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) -defined BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))))) is set
BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS n)) is non empty Relation-like BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL))
K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))),K233((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),K231((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),K232((R_VectorSpace_of_LinearOperators ((REAL-NS 1),(REAL-NS n))),(BoundedLinearOperators ((REAL-NS 1),(REAL-NS n)))),(BoundedLinearOperatorsNorm ((REAL-NS 1),(REAL-NS n))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1),(REAL-NS n))) is non empty set
(diff (g,x0)) . <*1*> is set
f is non empty Relation-like REAL -defined REAL 1 -valued Function-like total quasi_total V239() V240() V241() Element of K6(K7(REAL,(REAL 1)))
N1 is non empty Relation-like REAL -defined the carrier of (REAL-NS 1) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS 1)))
m * N1 is non empty Relation-like REAL -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7(REAL,REAL))
id REAL is non empty Relation-like REAL -defined REAL -valued Function-like one-to-one total quasi_total complex-valued ext-real-valued real-valued increasing non-decreasing Element of K6(K7(REAL,REAL))
g * N1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T * (id REAL) is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
f is V11() real ext-real Element of REAL
<*f*> is Relation-like NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued FinSequence-like FinSequence of REAL
[1,f] is set
{1,f} is non empty V160() V161() V162() set
{{1,f},{1}} is non empty set
{[1,f]} is non empty Function-like set
(diff (g,x0)) . <*f*> is set
f * (diff (T,S)) is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),f,(diff (T,S))) is set
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
m is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
m /. 0 is Element of the carrier of (REAL-NS n)
x0 is V11() real ext-real Element of REAL
S is V11() real ext-real Element of REAL
T is V11() real ext-real Element of REAL
|.T.| is V11() real ext-real Element of REAL
|.T.| " is V11() real ext-real Element of REAL
m /. T is Element of the carrier of (REAL-NS n)
||.(m /. T).|| is V11() real ext-real non negative Element of REAL
(|.T.| ") * ||.(m /. T).|| is V11() real ext-real Element of REAL
|.T.| * ((|.T.| ") * ||.(m /. T).||) is V11() real ext-real Element of REAL
|.T.| * x0 is V11() real ext-real Element of REAL
|.T.| * (|.T.| ") is V11() real ext-real Element of REAL
(|.T.| * (|.T.| ")) * ||.(m /. T).|| is V11() real ext-real Element of REAL
x0 * |.T.| is V11() real ext-real Element of REAL
1 * ||.(m /. T).|| is V11() real ext-real non negative Element of REAL
g is V11() real ext-real Element of REAL
g * |.T.| is V11() real ext-real Element of REAL
x0 * |.T.| is V11() real ext-real Element of REAL
m /. T is Element of the carrier of (REAL-NS n)
||.(m /. T).|| is V11() real ext-real non negative Element of REAL
m /. T is Element of the carrier of (REAL-NS n)
||.(m /. T).|| is V11() real ext-real non negative Element of REAL
x0 * |.T.| is V11() real ext-real Element of REAL
m /. T is Element of the carrier of (REAL-NS n)
||.(m /. T).|| is V11() real ext-real non negative Element of REAL
x0 * |.T.| is V11() real ext-real Element of REAL
T is V11() real ext-real Element of REAL
|.T.| is V11() real ext-real Element of REAL
m /. T is Element of the carrier of (REAL-NS n)
||.(m /. T).|| is V11() real ext-real non negative Element of REAL
x0 * |.T.| is V11() real ext-real Element of REAL
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
x0 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
S is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total V153( REAL-NS n, REAL-NS m) V154( REAL-NS n, REAL-NS m) Lipschitzian Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
S * x0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N1 is V11() real ext-real Element of REAL
dom S is Element of K6( the carrier of (REAL-NS n))
K6( the carrier of (REAL-NS n)) is set
rng x0 is Element of K6( the carrier of (REAL-NS n))
dom x0 is V160() V161() V162() Element of K6(REAL)
N2 is V11() real ext-real Element of REAL
N2 + N1 is V11() real ext-real Element of REAL
f is V11() real ext-real Element of REAL
f + N1 is V11() real ext-real Element of REAL
R2 is V11() real ext-real Element of REAL
R2 / 2 is V11() real ext-real Element of REAL
K39(2) is non empty V11() real ext-real positive non negative set
K37(R2,K39(2)) is V11() real ext-real set
1 + N1 is V11() real ext-real Element of REAL
(R2 / 2) / (1 + N1) is V11() real ext-real Element of REAL
K39((1 + N1)) is V11() real ext-real set
K37((R2 / 2),K39((1 + N1))) is V11() real ext-real set
0 / (1 + N1) is V11() real ext-real Element of REAL
K37(0,K39((1 + N1))) is V11() real ext-real set
R1 is V11() real ext-real Element of REAL
r is V11() real ext-real Element of REAL
|.r.| is V11() real ext-real Element of REAL
|.r.| " is V11() real ext-real Element of REAL
x0 /. r is Element of the carrier of (REAL-NS n)
||.(x0 /. r).|| is V11() real ext-real non negative Element of REAL
(|.r.| ") * ||.(x0 /. r).|| is V11() real ext-real Element of REAL
N1 + 1 is V11() real ext-real Element of REAL
(N1 + 1) * ((|.r.| ") * ||.(x0 /. r).||) is V11() real ext-real Element of REAL
(N1 + 1) * ((R2 / 2) / (1 + N1)) is V11() real ext-real Element of REAL
N1 * ||.(x0 /. r).|| is V11() real ext-real Element of REAL
(N1 + 1) * ||.(x0 /. r).|| is V11() real ext-real Element of REAL
S . (x0 /. r) is Element of the carrier of (REAL-NS m)
||.(S . (x0 /. r)).|| is V11() real ext-real non negative Element of REAL
(|.r.| ") * ||.(S . (x0 /. r)).|| is V11() real ext-real Element of REAL
(|.r.| ") * ((N1 + 1) * ||.(x0 /. r).||) is V11() real ext-real Element of REAL
S /. (x0 /. r) is Element of the carrier of (REAL-NS m)
(S * x0) /. r is Element of the carrier of (REAL-NS m)
||.((S * x0) /. r).|| is V11() real ext-real non negative Element of REAL
(|.r.| ") * ||.((S * x0) /. r).|| is V11() real ext-real Element of REAL
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
0. (REAL-NS m) is V68( REAL-NS m) Element of the carrier of (REAL-NS m)
the ZeroF of (REAL-NS m) is Element of the carrier of (REAL-NS m)
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T /. 0 is Element of the carrier of (REAL-NS n)
g is V11() real ext-real Element of REAL
N1 is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
N1 /. (0. (REAL-NS n)) is Element of the carrier of (REAL-NS m)
f is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
f + T is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
N1 * (f + T) is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N2 is Element of the carrier of (REAL-NS n)
||.N2.|| is V11() real ext-real non negative Element of REAL
dom N1 is Element of K6( the carrier of (REAL-NS n))
K6( the carrier of (REAL-NS n)) is set
rng (f + T) is Element of K6( the carrier of (REAL-NS n))
dom (f + T) is V160() V161() V162() Element of K6(REAL)
L2 is V11() real ext-real Element of REAL
L2 / 2 is V11() real ext-real Element of REAL
K39(2) is non empty V11() real ext-real positive non negative set
K37(L2,K39(2)) is V11() real ext-real set
1 + ||.N2.|| is non empty V11() real ext-real positive non negative Element of REAL
(L2 / 2) / (1 + ||.N2.||) is V11() real ext-real Element of REAL
K39((1 + ||.N2.||)) is non empty V11() real ext-real positive non negative set
K37((L2 / 2),K39((1 + ||.N2.||))) is V11() real ext-real set
0 / (1 + ||.N2.||) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() Element of REAL
K37(0,K39((1 + ||.N2.||))) is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V160() V161() V162() V163() V164() V165() V166() V233() V234() V235() V236() V237() V238() set
r is V11() real ext-real Element of REAL
r / (1 + ||.N2.||) is V11() real ext-real Element of REAL
K37(r,K39((1 + ||.N2.||))) is V11() real ext-real set
min (g,(r / (1 + ||.N2.||))) is V11() real ext-real Element of REAL
L0 is V11() real ext-real Element of REAL
|.L0.| is V11() real ext-real Element of REAL
T /. L0 is Element of the carrier of (REAL-NS n)
||.(T /. L0).|| is V11() real ext-real non negative Element of REAL
1 * |.L0.| is V11() real ext-real Element of REAL
f . L0 is Element of the carrier of (REAL-NS n)
L0 * N2 is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),L0,N2) is set
||.(f . L0).|| is V11() real ext-real non negative Element of REAL
||.N2.|| * |.L0.| is V11() real ext-real Element of REAL
||.(f . L0).|| - (||.N2.|| * |.L0.|) is V11() real ext-real Element of REAL
K38((||.N2.|| * |.L0.|)) is V11() real ext-real set
K36(||.(f . L0).||,K38((||.N2.|| * |.L0.|))) is V11() real ext-real set
(||.(f . L0).|| - (||.N2.|| * |.L0.|)) + (||.N2.|| * |.L0.|) is V11() real ext-real Element of REAL
N3 is V11() real ext-real Element of REAL
N3 + (||.N2.|| * |.L0.|) is V11() real ext-real Element of REAL
(f . L0) + (T /. L0) is Element of the carrier of (REAL-NS n)
||.((f . L0) + (T /. L0)).|| is V11() real ext-real non negative Element of REAL
||.(f . L0).|| + ||.(T /. L0).|| is V11() real ext-real non negative Element of REAL
(||.N2.|| * |.L0.|) + (1 * |.L0.|) is V11() real ext-real Element of REAL
||.N2.|| + 1 is non empty V11() real ext-real positive non negative Element of REAL
(||.N2.|| + 1) * |.L0.| is V11() real ext-real Element of REAL
(||.N2.|| + 1) * (r / (1 + ||.N2.||)) is V11() real ext-real Element of REAL
N1 /. ((f . L0) + (T /. L0)) is Element of the carrier of (REAL-NS m)
||.(N1 /. ((f . L0) + (T /. L0))).|| is V11() real ext-real non negative Element of REAL
((L2 / 2) / (1 + ||.N2.||)) * ||.((f . L0) + (T /. L0)).|| is V11() real ext-real Element of REAL
((L2 / 2) / (1 + ||.N2.||)) * ((||.N2.|| + 1) * |.L0.|) is V11() real ext-real Element of REAL
f /. L0 is Element of the carrier of (REAL-NS n)
(f /. L0) + (T /. L0) is Element of the carrier of (REAL-NS n)
N1 /. ((f /. L0) + (T /. L0)) is Element of the carrier of (REAL-NS m)
(f + T) /. L0 is Element of the carrier of (REAL-NS n)
N1 /. ((f + T) /. L0) is Element of the carrier of (REAL-NS m)
(N1 * (f + T)) /. L0 is Element of the carrier of (REAL-NS m)
|.L0.| " is V11() real ext-real Element of REAL
||.((N1 * (f + T)) /. L0).|| is V11() real ext-real non negative Element of REAL
(|.L0.| ") * ||.((N1 * (f + T)) /. L0).|| is V11() real ext-real Element of REAL
((L2 / 2) / (1 + ||.N2.||)) * (||.N2.|| + 1) is V11() real ext-real Element of REAL
(((L2 / 2) / (1 + ||.N2.||)) * (||.N2.|| + 1)) * |.L0.| is V11() real ext-real Element of REAL
(|.L0.| ") * ((((L2 / 2) / (1 + ||.N2.||)) * (||.N2.|| + 1)) * |.L0.|) is V11() real ext-real Element of REAL
|.L0.| * (|.L0.| ") is V11() real ext-real Element of REAL
(|.L0.| * (|.L0.| ")) * ((L2 / 2) / (1 + ||.N2.||)) is V11() real ext-real Element of REAL
((|.L0.| * (|.L0.| ")) * ((L2 / 2) / (1 + ||.N2.||))) * (||.N2.|| + 1) is V11() real ext-real Element of REAL
1 * ((L2 / 2) / (1 + ||.N2.||)) is V11() real ext-real Element of REAL
(1 * ((L2 / 2) / (1 + ||.N2.||))) * (||.N2.|| + 1) is V11() real ext-real Element of REAL
dom (N1 * (f + T)) is V160() V161() V162() Element of K6(REAL)
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
0. (REAL-NS m) is V68( REAL-NS m) Element of the carrier of (REAL-NS m)
the ZeroF of (REAL-NS m) is Element of the carrier of (REAL-NS m)
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
x0 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
x0 /. 0 is Element of the carrier of (REAL-NS n)
S is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
S /. (0. (REAL-NS n)) is Element of the carrier of (REAL-NS m)
T is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
T + x0 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
S * (T + x0) is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
g is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total V153( REAL-NS n, REAL-NS m) V154( REAL-NS n, REAL-NS m) Lipschitzian Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
g * x0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
(g * x0) + (S * (T + x0)) is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
n is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS n is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS n) is non empty non trivial set
K7(REAL, the carrier of (REAL-NS n)) is set
K6(K7(REAL, the carrier of (REAL-NS n))) is set
m is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V132() V159() V160() V161() V162() V163() V164() V165() Element of NAT
REAL-NS m is non empty non trivial V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() strict RealNormSpace-like V156() NORMSTR
the carrier of (REAL-NS m) is non empty non trivial set
K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is set
K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m))) is set
x0 is V11() real ext-real Element of REAL
g is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
g /. x0 is Element of the carrier of (REAL-NS n)
diff (g,x0) is Element of the carrier of (REAL-NS n)
dom g is V160() V161() V162() Element of K6(REAL)
N1 is V160() V161() V162() open Neighbourhood of x0
f is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
f * g is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
K7(REAL, the carrier of (REAL-NS m)) is set
K6(K7(REAL, the carrier of (REAL-NS m))) is set
diff ((f * g),x0) is Element of the carrier of (REAL-NS m)
diff (f,(g /. x0)) is Relation-like Function-like Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))
R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V87() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V122() V123() RealNormSpace-like V156() NORMSTR
BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) is non empty Element of K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))))
R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
LinearOperators ((REAL-NS n),(REAL-NS m)) is non empty functional Element of K6( the carrier of K284( the carrier of (REAL-NS n),(REAL-NS m)))
K284( the carrier of (REAL-NS n),(REAL-NS m)) is non empty V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) is non empty functional FUNCTION_DOMAIN of the carrier of (REAL-NS n), the carrier of (REAL-NS m)
FuncZero ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))
0. (REAL-NS m) is V68( REAL-NS m) Element of the carrier of (REAL-NS m)
the ZeroF of (REAL-NS m) is Element of the carrier of (REAL-NS m)
the carrier of (REAL-NS n) --> (0. (REAL-NS m)) is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7((Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))))
K7((Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is set
K7(K7((Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is set
K6(K7(K7((Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))) is set
FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m)) is Relation-like K7(REAL,(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) -defined Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))))
K7(REAL,(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is set
K7(K7(REAL,(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))) is set
K6(K7(K7(REAL,(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))),(Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))))) is set
RLSStruct(# (Funcs ( the carrier of (REAL-NS n), the carrier of (REAL-NS m))),(FuncZero ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncAdd ( the carrier of (REAL-NS n),(REAL-NS m))),(FuncExtMult ( the carrier of (REAL-NS n),(REAL-NS m))) #) is strict RLSStruct
the carrier of K284( the carrier of (REAL-NS n),(REAL-NS m)) is non empty set
K6( the carrier of K284( the carrier of (REAL-NS n),(REAL-NS m))) is set
K233(K284( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like Function-like Element of LinearOperators ((REAL-NS n),(REAL-NS m))
K231(K284( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((LinearOperators ((REAL-NS n),(REAL-NS m))),(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K232(K284( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) -defined LinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7(REAL,(LinearOperators ((REAL-NS n),(REAL-NS m)))),(LinearOperators ((REAL-NS n),(REAL-NS m))))) is set
RLSStruct(# (LinearOperators ((REAL-NS n),(REAL-NS m))),K233(K284( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K231(K284( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))),K232(K284( the carrier of (REAL-NS n),(REAL-NS m)),(LinearOperators ((REAL-NS n),(REAL-NS m)))) #) is V87() strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
K6( the carrier of (R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K233((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Element of BoundedLinearOperators ((REAL-NS n),(REAL-NS m))
K231((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is set
K232((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is Relation-like K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) -defined BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -valued Function-like total quasi_total Element of K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))))
K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))) is set
K6(K7(K7(REAL,(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m))))) is set
BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m)) is non empty Relation-like BoundedLinearOperators ((REAL-NS n),(REAL-NS m)) -defined REAL -valued Function-like total quasi_total complex-valued ext-real-valued real-valued Element of K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL))
K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL) is complex-valued ext-real-valued real-valued set
K6(K7((BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),REAL)) is set
NORMSTR(# (BoundedLinearOperators ((REAL-NS n),(REAL-NS m))),K233((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K231((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),K232((R_VectorSpace_of_LinearOperators ((REAL-NS n),(REAL-NS m))),(BoundedLinearOperators ((REAL-NS n),(REAL-NS m)))),(BoundedLinearOperatorsNorm ((REAL-NS n),(REAL-NS m))) #) is strict NORMSTR
the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS n),(REAL-NS m))) is non empty set
(diff (f,(g /. x0))) . (diff (g,x0)) is Element of the carrier of (REAL-NS m)
dom f is Element of K6( the carrier of (REAL-NS n))
K6( the carrier of (REAL-NS n)) is set
0. (REAL-NS n) is V68( REAL-NS n) Element of the carrier of (REAL-NS n)
the ZeroF of (REAL-NS n) is Element of the carrier of (REAL-NS n)
f /. (g /. x0) is Element of the carrier of (REAL-NS m)
N2 is Neighbourhood of g /. x0
R2 is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
R2 /. (0. (REAL-NS n)) is Element of the carrier of (REAL-NS m)
R2 is Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
R2 /. (0. (REAL-NS n)) is Element of the carrier of (REAL-NS m)
L1 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
L1 . 1 is Element of the carrier of (REAL-NS n)
R1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
L1 is non empty Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS n)))
L1 . 1 is Element of the carrier of (REAL-NS n)
R1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
r is Element of the carrier of (REAL-NS n)
(g /. x0) - (g /. x0) is Element of the carrier of (REAL-NS n)
- (g /. x0) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(g /. x0),(- (g /. x0))) is Element of the carrier of (REAL-NS n)
x0 - x0 is V11() real ext-real Element of REAL
K38(x0) is V11() real ext-real set
K36(x0,K38(x0)) is V11() real ext-real set
L1 . (x0 - x0) is Element of the carrier of (REAL-NS n)
R1 /. (x0 - x0) is Element of the carrier of (REAL-NS n)
(L1 . (x0 - x0)) + (R1 /. (x0 - x0)) is Element of the carrier of (REAL-NS n)
L1 . 0 is Element of the carrier of (REAL-NS n)
R1 /. 0 is Element of the carrier of (REAL-NS n)
(L1 . 0) + (R1 /. 0) is Element of the carrier of (REAL-NS n)
p0 is V11() real ext-real Element of REAL
p0 * r is Element of the carrier of (REAL-NS n)
the Mult of (REAL-NS n) is Relation-like K7(REAL, the carrier of (REAL-NS n)) -defined the carrier of (REAL-NS n) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)))
K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS n)), the carrier of (REAL-NS n))) is set
K466( the Mult of (REAL-NS n),p0,r) is set
(p0 * r) + (R1 /. 0) is Element of the carrier of (REAL-NS n)
(0. (REAL-NS n)) + (R1 /. 0) is Element of the carrier of (REAL-NS n)
L2 is non empty Relation-like the carrier of (REAL-NS n) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total V153( REAL-NS n, REAL-NS m) V154( REAL-NS n, REAL-NS m) Lipschitzian Element of K6(K7( the carrier of (REAL-NS n), the carrier of (REAL-NS m)))
L2 * L1 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7(REAL, the carrier of (REAL-NS m)))
dom (L2 * L1) is V160() V161() V162() Element of K6(REAL)
L2 . r is Element of the carrier of (REAL-NS m)
q is Element of the carrier of (REAL-NS m)
L0 is V11() real ext-real Element of REAL
(L2 * L1) . L0 is Element of the carrier of (REAL-NS m)
L0 * q is Element of the carrier of (REAL-NS m)
the Mult of (REAL-NS m) is Relation-like K7(REAL, the carrier of (REAL-NS m)) -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total Element of K6(K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m)))
K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m)) is set
K6(K7(K7(REAL, the carrier of (REAL-NS m)), the carrier of (REAL-NS m))) is set
K466( the Mult of (REAL-NS m),L0,q) is set
L1 . L0 is Element of the carrier of (REAL-NS n)
L2 . (L1 . L0) is Element of the carrier of (REAL-NS m)
L0 * r is Element of the carrier of (REAL-NS n)
K466( the Mult of (REAL-NS n),L0,r) is set
L2 . (L0 * r) is Element of the carrier of (REAL-NS m)
N3 is V160() V161() V162() open Neighbourhood of x0
g .: N3 is Element of K6( the carrier of (REAL-NS n))
L2 * R1 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
L1 + R1 is Relation-like REAL -defined the carrier of (REAL-NS n) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS n)))
R2 * (L1 + R1) is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
(L2 * R1) + (R2 * (L1 + R1)) is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
N is V160() V161() V162() open Neighbourhood of x0
dom (f * g) is V160() V161() V162() Element of K6(REAL)
x is set
x9 is V11() real ext-real Element of REAL
g . x9 is set
x is V11() real ext-real Element of REAL
g /. x is Element of the carrier of (REAL-NS n)
(g /. x) - (g /. x0) is Element of the carrier of (REAL-NS n)
K192((REAL-NS n),(g /. x),(- (g /. x0))) is Element of the carrier of (REAL-NS n)
x - x0 is V11() real ext-real Element of REAL
K36(x,K38(x0)) is V11() real ext-real set
L1 . (x - x0) is Element of the carrier of (REAL-NS n)
R1 /. (x - x0) is Element of the carrier of (REAL-NS n)
(L1 . (x - x0)) + (R1 /. (x - x0)) is Element of the carrier of (REAL-NS n)
g . x is set
dom R1 is V160() V161() V162() Element of K6(REAL)
dom L2 is Element of K6( the carrier of (REAL-NS n))
rng R1 is Element of K6( the carrier of (REAL-NS n))
dom (L2 * R1) is V160() V161() V162() Element of K6(REAL)
dom (L1 + R1) is V160() V161() V162() Element of K6(REAL)
dom R2 is Element of K6( the carrier of (REAL-NS n))
rng (L1 + R1) is Element of K6( the carrier of (REAL-NS n))
dom (R2 * (L1 + R1)) is V160() V161() V162() Element of K6(REAL)
dom ((L2 * R1) + (R2 * (L1 + R1))) is V160() V161() V162() Element of K6(REAL)
REAL /\ REAL is V160() V161() V162() set
L2 . (R1 /. (x - x0)) is Element of the carrier of (REAL-NS m)
L2 /. (R1 /. (x - x0)) is Element of the carrier of (REAL-NS m)
(L2 * R1) /. (x - x0) is Element of the carrier of (REAL-NS m)
R2 /. ((L1 . (x - x0)) + (R1 /. (x - x0))) is Element of the carrier of (REAL-NS m)
L1 /. (x - x0) is Element of the carrier of (REAL-NS n)
(L1 /. (x - x0)) + (R1 /. (x - x0)) is Element of the carrier of (REAL-NS n)
R2 /. ((L1 /. (x - x0)) + (R1 /. (x - x0))) is Element of the carrier of (REAL-NS m)
(L1 + R1) /. (x - x0) is Element of the carrier of (REAL-NS n)
R2 /. ((L1 + R1) /. (x - x0)) is Element of the carrier of (REAL-NS m)
(R2 * (L1 + R1)) /. (x - x0) is Element of the carrier of (REAL-NS m)
(L2 * L1) . (x - x0) is Element of the carrier of (REAL-NS m)
L2 . (L1 . (x - x0)) is Element of the carrier of (REAL-NS m)
(f * g) /. x is Element of the carrier of (REAL-NS m)
(f * g) /. x0 is Element of the carrier of (REAL-NS m)
((f * g) /. x) - ((f * g) /. x0) is Element of the carrier of (REAL-NS m)
- ((f * g) /. x0) is Element of the carrier of (REAL-NS m)
K192((REAL-NS m),((f * g) /. x),(- ((f * g) /. x0))) is Element of the carrier of (REAL-NS m)
f /. (g /. x) is Element of the carrier of (REAL-NS m)
(f /. (g /. x)) - ((f * g) /. x0) is Element of the carrier of (REAL-NS m)
K192((REAL-NS m),(f /. (g /. x)),(- ((f * g) /. x0))) is Element of the carrier of (REAL-NS m)
(f /. (g /. x)) - (f /. (g /. x0)) is Element of the carrier of (REAL-NS m)
- (f /. (g /. x0)) is Element of the carrier of (REAL-NS m)
K192((REAL-NS m),(f /. (g /. x)),(- (f /. (g /. x0)))) is Element of the carrier of (REAL-NS m)
(diff (f,(g /. x0))) . ((g /. x) - (g /. x0)) is Element of the carrier of (REAL-NS m)
R2 /. ((g /. x) - (g /. x0)) is Element of the carrier of (REAL-NS m)
((diff (f,(g /. x0))) . ((g /. x) - (g /. x0))) + (R2 /. ((g /. x) - (g /. x0))) is Element of the carrier of (REAL-NS m)
(L2 . (L1 . (x - x0))) + (L2 . (R1 /. (x - x0))) is Element of the carrier of (REAL-NS m)
((L2 . (L1 . (x - x0))) + (L2 . (R1 /. (x - x0)))) + ((R2 * (L1 + R1)) /. (x - x0)) is Element of the carrier of (REAL-NS m)
((L2 * R1) /. (x - x0)) + ((R2 * (L1 + R1)) /. (x - x0)) is Element of the carrier of (REAL-NS m)
(L2 . (L1 . (x - x0))) + (((L2 * R1) /. (x - x0)) + ((R2 * (L1 + R1)) /. (x - x0))) is Element of the carrier of (REAL-NS m)
L0 is non empty Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like total quasi_total linear Element of K6(K7(REAL, the carrier of (REAL-NS m)))
L0 . (x - x0) is Element of the carrier of (REAL-NS m)
R0 is Relation-like REAL -defined the carrier of (REAL-NS m) -valued Function-like RestFunc-like Element of K6(K7(REAL, the carrier of (REAL-NS m)))
R0 /. (x - x0) is Element of the carrier of (REAL-NS m)
(L0 . (x - x0)) + (R0 /. (x - x0)) is Element of the carrier of (REAL-NS m)
(L2 * L1) . 1 is Element of the carrier of (REAL-NS m)